Interferometer and method for measuring the refractive index and optical path length effects of air

ABSTRACT

Apparatus and methods particularly suitable for use in electro-optical metrology and other applications to measure and monitor the refractive index of a gas in a measurement path and/or the change in optical path length of the measurement path due to the gas while the refractive index of the gas may be fluctuating due to turbulence or the like and/or the physical length of the measuring path may be changing. More specifically, the invention employs electronic frequency processing to provide measurements of dispersion of the refractive index, the dispersion being substantially proportional to the density of the gas, and/or measurements of dispersion of the optical path length, the dispersion of the optical path length being related to the dispersion of the refractive index and the physical length of the measurement path. The refractive index of the gas and/or the optical path length effects of the gas are subsequently computed from the measured dispersion of the refractive index and/or the measured dispersion of the optical path length, respectively. The information generated by the inventive apparatus is particularly suitable for use in interferometric distance measuring instruments (DMI) to compensate for errors related to refractive index of gas in a measurement path brought about by environmental effects and turbulence induced by rapid stage slew rates. In preferred embodiments, differential plane mirror interferometer architectures are utilized, the operating wavelengths are approximately harmonically related and may be monitored and/or controlled to meet precision requirements, heterodyne and superheterodyne processing are beneficially used, and phase redundancy is resolved.

CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority from U.S. ProvisionalApplication No. 60/075,586 filed on Feb. 23, 1998 and entitled“INTERFEROMETER AND METHOD FOR MEASURING THE REFRACTIVE INDEX ANDOPTICAL PATH LENGTH EFFECTS OF AIR” and is a continuation-in-part ofU.S. patent application Ser. No. 09/078,254 filed on May 13, 1998 andentitled “INTERFEROMETRIC APPARATUS AND METHODS USING ELECTRONICFREQUENCY PROCESSING FOR MEASURING AND COMPENSATING FOR REFRACTIVE INDEXEFFECTS IN AN OPTICAL PATH”. Both of said applications are commonlyowned herewith, and their contents are incorporated herein by reference.This application is also related to commonly owned U.S. Pat. No.5,838,485.

FIELD OF THE INVENTION

[0002] The present invention relates to optical instruments formeasuring distance and refractive index. The invention relates inparticular to interferometric distance measurement independent of theoptical path length effects of refractive index of gas in a measurementpath including the effects of refractive index fluctuations.

BACKGROUND AND PRIOR ART

[0003] A frequently-encountered problem in metrology is the measurementof the refractive index of a column of air. Several techniques exist formeasuring the index under highly controlled circumstances, such as whenthe air column is contained in a sample cell and is monitored fortemperature, pressure, and physical dimension. See for example, anarticle entitled “An air refractometer for interference lengthmetrology,” by J. Terrien, Metrologia 1(3), 80-83 (1965).

[0004] Perhaps the most difficult measurement related to the refractiveindex of air is the measurement of refractive index fluctuations over ameasurement path of unknown or variable length, with uncontrolledtemperature and pressure. Such circumstances arise frequently ingeophysical and meteorological surveying, for which the atmosphere isobviously uncontrolled and the refractive index is changing dramaticallybecause of variations in air density and composition. The problem isdescribed in an article entitled “Effects of the atmospheric phasefluctuation on long-distance measurement,” by H. Matsumoto and K.Tsukahara, Appl. Opt. 23(19), 3388-3394 (1984), and in an articleentitled “Optical path length fluctuation in the atmosphere,” by G. N.Gibson et al., Appl. Opt. 23(23), 4383-4389 (1984).

[0005] Another example situation is high-precision distance measuringinterferometry, such as is employed in micro-lithographic fabrication ofintegrated circuits. See for example an article entitled “Residualerrors in laser interferometry from air turbulence and nonlinearity,” byN. Bobroff, Appl. Opt. 26(13), 2676-2682 (1987), and an article entitled“Recent advances in displacement measuring interferometry,” also by N.Bobroff, Measurement Science & Tech. 4(9), 907-926 (1993). As noted inthe aforementioned cited references, interferometric displacementmeasurements in air are subject to environmental uncertainties,particularly to changes in air pressure and temperature; touncertainties in air composition such as resulting from changes inhumidity; and to the effects of turbulence in the air. Such factorsalter the wavelength of the light used to measure the displacement.Under normal conditions the refractive index of air is approximately1.0003 with a variation of the order of 1×10⁻⁵ to 1×10⁻⁴. In manyapplications the refractive index of air must be known with a relativeprecision of less than 0.1 ppm (parts per million) to 0.003 ppm, thesetwo relative precisions corresponding to a displacement measurementaccuracy of 100 nm and 3 nm, respectively, for a one meterinterferometric displacement measurement.

[0006] There are frequent references in the art to heterodyne methods ofphase estimation, in which the phase varies with time in a controlledway. For example, in a known form of prior-art heterodynedistance-measuring interferometer, the source emits two orthogonallypolarized beams having slightly different optical frequencies (e.g. 2MHz). The interferometric receiver in this case is typically comprisedof a linear polarizer and a photodetector to measure a time-varyinginterference signal. The signal oscillates at the beat frequency and thephase of the signal corresponds to the relative phase difference. Afurther representative example of the prior art in heterodynedistance-measuring interferometry is taught in commonly-owned U.S. Pat.No. 4,688,940 issued to G. E. Sommargren and M. Schaham (1987). However,these known forms of interferometric metrology are limited byfluctuations in refractive index, and by themselves are unsuited to thenext generation of microlithography instruments.

[0007] Another known form of interferometer for distance measurement isdisclosed in U.S. Pat. No. 4,005,936 entitled “Interferometric MethodsAnd Apparatus For Measuring Distance To A Surface” issued to J. D.Redman and M. R. Wall (1977). The method taught by Redman and Wallconsists of employing laser beams of two different wavelengths, each ofwhich is split into two parts. Frequency shifts are introduced into onepart of the respective beams. One part of each beam reflects from anobject and recombines with the other part on a photodetector. From theinterference signal at the detector is derived a phase, at a differencefrequency, that is a measure of the distance to the surface. Theequivalent wavelength of the phase associated with the differencefrequency is equal to the product of the two laser wavelengths dividedby the difference of the two wavelengths. This two-wavelength techniqueof Redman and Wall reduces measurement ambiguities, but is at least assensitive to the deleterious effects of refractive index fluctuations ofthe air as single-wavelength techniques.

[0008] Another example of a two-wavelength interferometer similar tothat of Redman and Wall is disclosed in U.S. Pat. No. 4,907,886 entitled“Method And Apparatus For Two-Wavelength Interferometry With OpticalHeterodyne Processes And Use For Position Or Range Finding,” issued toR. Dändliker and W. Heerburgg (1990). This system is also described inan article entitled “Two-Wavelength Laser Interferometry UsingSuperheterodyne Detection,” by R. Dändliker, R. Thalmann, and D.Prongué, Opt. Let. 13(5), 339-341 (1988), and in an article entitled“High-Accuracy Distance Measurements With Multiple-WavelengthInterferometry,” by R. Dändliker, K. Hug, J. Politch, and E. Zimmermann.The system of Dändliker et al., as taught in U.S. Pat. No. 4,907,886,employs laser beams of two wavelengths, each of the beams comprising twopolarization components separated in frequency by means of acousto-opticmodulation. After passing these beams collinearly through a Michelsoninterferometer, the polarization components are mixed, resulting in aninterference signal, i.e. a heterodyne signal. In that the heterodynesignal has a different frequency for each of the two wavelengths, aso-called superheterodyne signal results therefrom having a frequencyequal to the difference in the heterodyne frequencies and a phaseassociated with an equivalent wavelength equal to the product of the twolaser wavelengths divided by the difference of the two wavelengths.According to U.S. Pat. No. 4,907,886 (cited above), the phase of thesuperheterodyne signal is assumed to be dependent only on the positionof a measurement object and the equivalent wavelength. Therefore, thissystem is also not designed to measure or compensate for thefluctuations in the refractive index of air.

[0009] Further examples of the two-wavelength superheterodyne techniquedeveloped by Redman and Wall and by Dändliker and Heerburgg (citedabove) are found in an article entitled “Two-wavelength doubleheterodyne interferometry using a matched grating technique,” by Z.Sodnik, E. Fischer, T. Ittner, and H. J. Tiziani, Appl. Opt. 30(22),3139-3144 (1991), and in an article entitled “Diode laser and fiberoptics for dual-wavelength heterodyne interferometry,” by S. Manhart andR. Maurer, SPIE 1319, 214-216 (1990). However, neither one of theseexamples addresses the problem of refractive index fluctuations.

[0010] It may be concluded from the foregoing that the prior art inheterodyne and superheterodyne interferometry does not provide a highspeed method and corresponding means for measuring and compensating theoptical path length effects of air in a measuring path, particularlyeffects due to fluctuations in the refractive index of air. Thisdeficiency in the prior art results in significant measurementuncertainty, thus seriously affecting the precision of systems employingsuch interferometers as found for example in micro-lithographicfabrication of integrated circuits. Future interferometers willnecessarily incorporate an inventive, new method and means for measuringand compensating a fluctuating refractive index in a measurement pathcomprised of a changing physical length.

[0011] One way to detect refractive index fluctuations is to measurechanges in pressure and temperature along a measurement path andcalculate the effect on the optical path length of the measurement path.Mathematical equations for effecting this calculation are disclosed inan article entitled “The Refractivity Of Air,” by F. E. Jones, J. Res.NBS 86(1), 27-32 (1981). An implementation of the technique is describedin an article entitled “High-Accuracy Displacement Interferometry InAir,” by W. T. Estler, Appl. Opt. 24(6), 808-815 (1985). Unfortunately,this technique provides only approximate values, is cumbersome, andcorrects only for slow, global fluctuations in air density.

[0012] Another, more direct way to detect the effects of a fluctuatingrefractive index over a measurement path is by multiple-wavelengthdistance measurement. The basic principle may be understood as follows.Interferometers and laser radar measure the optical path length betweena reference and an object, most often in open air. The optical pathlength is the integrated product of the refractive index and thephysical path traversed by a measurement beam. In that the refractiveindex varies with wavelength, but the physical path is independent ofwavelength, it is generally possible to determine the physical pathlength from the optical path length, particularly the contributions offluctuations in refractive index, provided that the instrument employsat least two wavelengths. The variation of refractive index withwavelength is known in the art as dispersion, therefore this techniquewill be referred to hereinafter as the dispersion technique.

[0013] The dispersion technique for refractive index measurement has along history, and predates the introduction of the laser. An articleentitled “Long-Path Interferometry Through An Uncontrolled Atmosphere,”by K. E. Erickson, JOSA 52(7), 781-787 (1962), describes the basicprinciples and provides an analysis of the feasibility of the techniquefor geophysical measurements. Additional theoretical proposals are foundin an article entitled “Correction Of Optical Distance Measurements ForThe Fluctuating Atmospheric Index Of Refraction,” by P. L. Bender and J.C. Owens, J. Geo. Res. 70(10), 2461-2462 (1965).

[0014] Commercial distance-measuring laser radar based on the dispersiontechnique for refractive index compensation appeared in the 1970's. Anarticle entitled “Two-Laser Optical Distance-Measuring Instrument ThatCorrects For The Atmospheric Index Of Refraction,” by K. B. Earnshaw andE. N. Hernandez, Appl. Opt. 11(4), 749-754 (1972), discloses aninstrument employing microwave-modulated HeNe and HeCd lasers foroperation over a 5 to 10 km measurement path. Further details of thisinstrument are found in an article entitled “Field Tests Of A Two-Laser(4416A and 6328A) Optical Distance-Measuring Instrument Correcting ForThe Atmospheric Index Of Refraction,” by E. N. Hernandez and K. B.Earnshaw, J. Geo. Res. 77(35), 6994-6998 (1972). Further examples ofapplications of the dispersion technique are discussed in an articleentitled “Distance Corrections For Single-And Dual-Color Lasers By RayTracing,” by E. Berg and J. A. Carter, J. Geo. Res. 85(B11), 6513-6520(1980), and in an article entitled “A Multi-WavelengthDistance-Measuring Instrument For Geophysical Experiments,” by L. E.Slater and G. R. Huggett, J. Geo. Res. 81(35), 6299-6306 (1976).

[0015] Although instrumentation for geophysical measurements typicallyemploys intensity-modulation laser radar, it is understood in the artthat optical interference phase detection is more advantageous forshorter distances. In U.S. Pat. No. 3,647,302 issued in 1972 to R. B.Zipin and J. T. Zalusky, entitled “Apparatus For And Method Of ObtainingPrecision Dimensional Measurements,” there is disclosed aninterferometric displacement-measuring system employing multiplewavelengths to compensate for variations in ambient conditions such astemperature, pressure, and humidity. The instrument is specificallydesigned for operation with a movable object, that is, with a variablephysical path length. However, the phase-detection means of Zipin andZalusky is insufficiently accurate for high-precision measurement.

[0016] A more modern and detailed example is the system described in anarticle by Y. Zhu, H. Matsumoto, T. O'ishi, SPIE 1319, Optics in ComplexSystems, 538-539 (1990), entitled “Long-Arm Two-Color Interferometer ForMeasuring The Change Of Air Refractive Index.” The system of Zhu et al.employs a 1064 nm wavelength YAG laser and an 632 nm HeNe laser togetherwith quadrature phase detection. Substantially the same instrument isdescribed in Japanese in an earlier article by Zhu et al. entitled“Measurement Of Atmospheric Phase And Intensity Turbulence For Long-PathDistance Interferometer,” Proc. 3^(rd) Meeting On Lightwave SensingTechnology, Appl. Phys. Soc. of Japan, 39 (1989). However, theinterferometer of Zhu et al. has insufficient resolution for allapplications, e.g. sub-micron interferometry for microlithography.

[0017] A recent attempt at high-precision interferometry formicrolithography is represented by U.S. Pat. No. 4,948,254 issued to A.Ishida (1990). A similar device is described by Ishida in an articleentitled “Two Wavelength Displacement-Measuring Interferometer UsingSecond-Harmonic Light To Eliminate Air-Turbulence-Induced Errors,” Jpn.J. Appl. Phys. 28(3), L473-475 (1989). In the article, adisplacement-measuring interferometer is disclosed which eliminateserrors caused by fluctuations in the refractive index by means oftwo-wavelength dispersion detection. An Ar⁺ laser source provides bothwavelengths simultaneously by means of a frequency-doubling crystalknown in the art as BBO. The use of a BBO doubling crystal results intwo wavelengths that are fundamentally phase locked, thus greatlyimproving the stability and accuracy of the refractive indexmeasurement. However, the phase detection means, which employ simplehomodyne quadrature detection, are insufficient for high resolutionphase measurement. Further, the phase detection and signal processingmeans are not suitable for dynamic measurements, in which the motion ofthe object results in rapid variations in phase that are difficult todetect accurately.

[0018] In U.S. Pat. No. 5,404,222 entitled “Interferometric MeasuringSystem With Air Turbulence Compensation,” issued to S. A. Lis (1995),there is disclosed a two-wavelength interferometer employing thedispersion technique for detecting and compensating refractive indexfluctuations. A similar device is described by Lis in an articleentitled “An Air Turbulence Compensated Interferometer For ICManufacturing,” SPIE 2440 (1995). Improvement on U.S. Pat. No. 5,404,222by S. A. Lis is disclosed in U.S. Pat. No. 5,537,209, issued July 1996.The principal innovation of this system with respect to that taught byIshida in Jpn. J. Appl. Phys. (cited above) is the addition of a secondBBO doubling crystal to improve the precision of the phase detectionmeans. The additional BBO crystal makes it possible to opticallyinterfere two beams having wavelengths that are exactly a factor of twodifferent. The resultant interference has a phase that is directlydependent on the refractive index but is substantially independent ofstage motion. However, the system taught by Lis has the disadvantagethat it is complicated and requires an additional BBO crystal for everymeasurement path. In that microlithography stages frequently involve sixor more measurement paths, and that BBO can be relatively expensive, theadditional crystals are a significant cost burden. An additionaldisadvantage of Lis' system is that it employs a low-speed (32-Hz) phasedetection system based on the physical displacement of a PZT transducer.

[0019] It is clear from the foregoing, that the prior art does notprovide a practical, high-speed, high-precision method and correspondingmeans for measuring refractive index of air and measuring andcompensating for the optical path length effects of the air in ameasuring path, particularly the effects due to fluctuations in therefractive index of the air. The limitations in the prior art ariseprincipally from the following unresolved technical difficulties: (1)Prior-art heterodyne and superheterodyne interferometers are limited inaccuracy by fluctuations in the refractive index of air; (2) Prior-artdispersion techniques for measuring index fluctuations require extremelyhigh accuracy in interference phase measurement, typically exceeding byan order of magnitude the typical accuracy of high-precisiondistance-measuring interferometers; (3) Obvious modifications toprior-art interferometers to improve phase-measuring accuracy wouldincrease the measurement time to an extent incompatible with therapidity of stage motion in modern microlithography equipment; (4)Prior-art dispersion techniques require at least two extremely stablelaser sources, or a single source emitting multiple, phase-lockedwavelengths; (5) Prior-art dispersion techniques in microlithographyapplications are sensitive to stage motion during the measurement,resulting in systematic errors; and (6) Prior-art dispersion techniquesthat employ doubling crystals (e.g. U.S. Pat. No. 5,404,222 to Lis) aspart of the detection system are expensive and complicated.

[0020] These deficiencies in the prior art have led to the absence ofany practical interferometric system for performing displacementmeasurement for microlithography in the presence of a gas in ameasurement path where there are typically refractive index fluctuationsand the measurement path is comprised of a changing physical length.

[0021] Accordingly, it is an object of the invention to provide a methodand apparatus for rapidly and accurately measuring and monitoring therefractive index of a gas in a measurement path and/or the optical pathlength effects of the gas wherein the refractive index may befluctuating and/or the physical length of the measurement path may bechanging.

[0022] It is another object of the invention to provide a method andapparatus for rapidly and accurately measuring and monitoring therefractive index of a gas in a measurement path and/or the optical pathlength effects of the gas wherein the accuracy of measurements andmonitoring of the refractive index of the gas and/or of the optical pathlength effects of the gas are substantially not compromised by a rapidchange in physical length of measurement path.

[0023] It is another object of the invention to provide a method andapparatus for rapidly and accurately measuring and monitoring therefractive index of a gas in a measurement path and/or the optical pathlength effects of the gas wherein the method and apparatus does notrequire measurement and monitoring of environmental conditions such astemperature and pressure.

[0024] It is another object of the invention to provide a method andapparatus for rapidly and accurately measuring and monitoring therefractive index of a gas in a measurement path and/or the optical pathlength effects of the gas wherein the method and apparatus may use butdoes not require the use of two or more optical beams of differingwavelengths which are phase locked.

[0025] It is another object of the invention to provide a method andapparatus for rapidly and accurately measuring and monitoring theoptical path length effects of a gas in a measurement path wherein thelengths of measuring paths in an interferometric measurement aresubstantially not used in a computation of the optical path lengtheffects of the gas.

[0026] Other objects of the invention will, in part, be obvious andwill, in part, appear hereinafter. The invention accordingly comprisesmethods and apparatus possessing the construction, steps, combination ofelements, and arrangement of parts exemplified in the detaileddescription to follow when read in connection with the drawings.

SUMMARY OF THE INVENTION

[0027] The present invention generally relates to apparatus and methodsfor measuring and monitoring the refractive index of a gas in ameasurement path and/or the change in optical path length of themeasurement path due to the gas wherein the refractive index of the gasmay be fluctuating, e.g., the gas is turbulent, and/or the physicallength of the measuring path may be changing. The present invention alsorelates to apparatus and methods for use in electro-optical metrologyand other applications. More specifically, the invention operates toprovide measurements of dispersion of the refractive index, thedispersion being substantially proportional to the density of the gas,and/or measurements of dispersion of the optical path length, thedispersion of the optical path length being related to the dispersion ofthe refractive index and the physical length of the measurement path.The refractive index of the gas and/or the optical path length effectsof the gas are subsequently computed from the measured dispersion of therefractive index and/or the measured dispersion of the optical pathlength, respectively. The information generated by the inventiveapparatus is particularly suitable for use in interferometric distancemeasuring instruments (DMI) to compensate for errors related torefractive index of gas in a measurement path brought about byenvironmental effects and turbulence induced by rapid stage slew rates.

[0028] Several embodiments of the invention have been made and thesefall broadly into two categories that address the need for more or lessprecision in final measurements. While the various embodiments sharecommon features, they differ in some details to achieve individualgoals.

[0029] In general, the inventive apparatus comprises interferometermeans having first and second measurement legs at least one of whichchanges in length and at least one of which is at least in part occupiedby the gas. Perferably a reference leg and a measurement leg are used inpreferred embodiments. The constituent legs are preferably configuredand arranged so that the measurement leg has a portion of its opticalpath length substantially the same as the optical path length of thereference leg. The gas in the remaining portion of the optical path ofthe measurement leg in a typical interferometric DMI application is air.

[0030] Means for generating at least two light beams having differentwavelengths are included. In preferred embodiments, a source generates aset of light beams, the set of light beams being comprised of at leasttwo light beams, each beam of the set of light beams having a differentwavelength. The relationship between the wavelengths of the beams of theset of light beams, the approximate relationship, is known.

[0031] A set of frequency-shifted light beams is generated from the setof light beams by introducing a frequency difference between twoorthogonally polarized components of each beam of the set of light beamssuch that no two beams of the set of frequency-shifted light beams havethe same frequency difference. For a given embodiment, the ratios of thewavelengths are the same as the known approximate relationship torelative precisions which depend on chosen operating wavelenghs and thecorresponding known approximate relationship. Because of this wavelengthdependence, these relative precisions are referred to as the respectiverelative precisions of the ratios of the wavelengths. In a number ofembodiments, the respective relative precisions of the ratios of thewavelengths are of an order of magnitude less than the respectivedispersions of the gas times the relative precision required for themeasurement of the respective refractive indices of the gas and/or forthe measurement of the respective changes in the optical path length ofthe measurement leg due to the gas.

[0032] In certain ones of the embodiments, the approximate relationshipis expressed as a sequence of ratios, each ratio comprising a ratio oflow order non-zero integers, e.g., 2/1, to respective relativeprecisions, the respective relative precisions of the sequence ofratios, wherein a respective relative precision of the respectiverelative precisions of the sequence of ratios is of an order ofmagnitude less than the respective dispersion of the gas times therespective relative precision required for the measurement of therespective refractive index of the gas and/or for the measurement of therespective change in the optical path length of the measurement leg dueto the gas.

[0033] In other embodiments, where the respective relative precisions ofthe ratios of the wavelengths is inappropriate to the desired value,means are provided for monitoring the ratios of the wavelengths andeither providing feedback to control the respective relative precisionsof the ratios of the wavelengths, information to correct subsequentcalculations influenced by undesirable departures of the respectiverelative precisions of the ratios of the wavelengths from the desiredrespective relative precisions of the ratios of the wavelengths, or somecombination of both. Means are also provided for monitoring thewavelength used in the primary objective of DMI, the determination of achange in a length of the measurement path.

[0034] At least a portion of each of the frequency-shifted light beamsis introduced into the interferometer means by suitable optical means sothat a first portion of at least a portion of each frequency-shiftedlight beam travels through the reference leg along predetermined pathsof the reference leg and a second portion of at least a portion of eachfrequency-shifted light beam travels through the measurement leg alongpredetermined paths of the measurement leg, the first and secondportions of at least a portion of each frequency-shifted light beambeing different. Afterwards, the first and second portions of at least aportion of each frequency-shifted light beam emerge from theinterferometer means as exit beams containing information about theoptical path length through the predetermined paths in the reference legand the optical path length through the predetermined paths in themeasurement leg.

[0035] Combining means are provided for receiving the exit beams toproduce mixed optical signals which contain information corresponding tothe phase differences between the exit beams of the first and secondportions of at least a portion of each frequency-shifted light beam. Themixed optical signals are then sensed by a photodetector, preferably byphotoelectric detection, which operates to generate electricalinterference signals that contain information corresponding to therefractive index of the gas at the different beam wavelengths and to theoptical path length in the measurement leg due to the refractive indexof the gas at the different beam wavelengths.

[0036] In certain of the embodiments, modified electrical interferencesignals are then generated from the electrical interference signals byeither multiplying or dividing the phase of each of the electricalinterference signals by a number, the relationship of the numbers beingeither the same as the known approximate relationship of the wavelengthsor the same as the reciprocal of the known approximate relationship ofthe wavelengths, respectively.

[0037] The electrical interference signals, or the correspondingmodified electrical interference signals depending on the embodiment,are then analyzed by electronic means that operate to determine thedispersion of the optical path length of the measurement legsubstantially due to the dispersion of the refractive index of the gasand/or the dispersion (n_(i)-n_(j)) of the gas where i and j areintegers corresponding to wavelengths and different from one another.From this information and the reciprocal dispersive power of the gas,the refractivity of the gas, (n_(r)-1) where r is an integercorresponding to a wavelength, and/or the contribution to the opticalpath length of the measurement leg due to the refractive index of thegas can also be determined by the electronic means. The value of r maybe different from i and j or equal to either i or j. The electronicmeans can comprise electronic means in the form of a microprocessor or ageneral purpose computer suitably programmed in well-known ways toperform the needed calculations.

[0038] In preferred form, the electrical interference signals compriseheterodyne signals containing phase information corresponding to therefractive index of the gas and to the optical path length of themeasurement leg and the apparatus further comprises means to determinethe phases of the heterodyne signals to generate phase informationcorresponding to the dispersion of the refractive index of the gas andto the dispersion of the optical path length of the measurement leg dueto the dispersion of the refractive index of the gas. In certain of theembodiments, the apparatus further comprises means for mixing, i.e.multiplying, the modified heterodyne signals corresponding to themodified electrical signals to generate at least one modifiedsuperheterodyne signal containing phase corresponding to the dispersionof the refractive index of the gas and to the dispersion of the opticalpath length of the measurement leg due to the dispersion of therefractive index of the gas. Means are also included for resolving phaseambiguities of the heterodyne signals, modified heterodyne signals, andthe modified superheterodyne signals generated in certain of theembodiments. Depending on the details of the optical paths experiencedby the light beam portions as they travel through the interferometermeans of the various embodiments, additional or different electronicsare provided.

[0039] While the inventive method disclosed may be carried out using thepreferred apparatus described, it will be evident that it may also bepracticed using other well-known apparatus. In addition, it is shownthat apparatus may be employed which uses homodyne signals.

BRIEF DESCRIPTION OF THE DRAWINGS

[0040] The structure and operation of the invention, together with otherobjects and advantages thereof, may best be understood by reading thedetailed description in conjunction with the drawings wherein theinvention's parts have an assigned reference numeral that is used toidentify them in all of the drawings in which they appear and wherein:

[0041]FIGS. 1a-1 b taken together illustrate, in diagrammatic form, thepresently first preferred embodiment of the present invention with

[0042]FIG. 1a showing optical paths between indicated elements source 1,modulator 3, source 2, modulator 4, interferometer 260, detectors 85 and86, and translator 267 and the paths of electrical signals betweenindicated elements driver 5, modulator 3, driver 6, modulator 4,detectors 85 and 86, electronic processor 109, and computer 110;

[0043]FIG. 1b is a drawing showing a block diagram of the processingelectronics 109;

[0044]FIG. 1c is a drawing showing a block diagram of the processingelectronics 109A for the first variant of the first embodiment;

[0045]FIG. 1d is a drawing showing a block diagram of the processingelectronics 109B for the third variant of the first embodiment;

[0046]FIGS. 2a-2 f taken together illustrate, in diagrammatic form, thepresently second preferred embodiment of the present invention with

[0047]FIG. 2a showing optical paths between indicated elements source 1,modulator 3, source 2, modulator 4, differential plane mirrorinterferometers 69 and 70, beam splitter 65, external mirror system 90,detectors 185 and 186, and translator 67 and the paths of electricalsignals between indicated elements driver 5, modulator 3, driver 6,modulator 4, detectors 185 and 186, electronic processor 209, andcomputer 110;

[0048]FIG. 2b illustrates differential plane mirror interferometer 69;

[0049]FIG. 2c illustrates differential plane mirror interferometer 70;

[0050]FIG. 2d illustrates external mirror system 90, furnishing theexternal mirrors for differential plane mirror interferometer 69, andstage translator 67;

[0051]FIG. 2e illustrates external mirror system 90, furnishing theexternal mirrors for differential plane mirror interferometer 70, andstage translator 67;

[0052]FIG. 2f is a drawing showing a block diagram of the processingelectronics 209;

[0053]FIGS. 3a-3 b taken together illustrate, in diagrammatic form, thepresently preferred third embodiment of the present invention with

[0054]FIG. 3a showing optical paths and electronic paths of apparatusfor determination of refractive index of a gas and/or the optical pathlength effects of the gas comprised in part of the same apparatus as forthe first preferred embodiment and optical paths and electronic paths ofapparatus for determination of χ and the ratio K/χ, a number of elementsof the apparatus for determination of χ and the ratio K/χ performinganalogous operations as apparatus of the first preferred embodimentapart from the suffix “b” when referring to apparatus for determinationof χ and the ratio K/χ;

[0055]FIG. 3b is a drawing showing a block diagram of the processingelectronics 109 b;

[0056]FIGS. 4a-4 c taken together illustrate, in diagrammatic form, thepresently preferred fourth embodiment of the present invention with

[0057]FIG. 4a showing optical paths and electronic paths of apparatusfor determination of refractive index of a gas and/or the optical pathlength effects of the gas comprised in part of the same apparatus as forthe second preferred embodiment and optical paths and electronic pathsof apparatus for determination of the χ and ratio K/χ, a number ofelements of the apparatus for determination of χ and the ratio K/χperforming analogous operations as apparatus of the second preferredembodiment apart from the suffix “b” when referring to apparatus fordetermination of X and the ratio K/χ;

[0058]FIG. 4b illustrates the external mirror system 90 b furnishing theexternal mirrors for differential plane mirror interferometer 69 b;

[0059]FIG. 4c illustrates the external mirror system 90 b furnishing theexternal mirrors for differential plane mirror interferometer 70 b;

[0060]FIG. 5 is a high-level flowchart depicting various steps carriedout in practicing a method in accordance with the invention;

[0061]FIGS. 6a-6 c relate to lithography and its application tomanufacturing integrated circuits wherein

[0062]FIG. 6a is a schematic drawing of a lithography exposure systememploying the interferometry system.

[0063]FIGS. 6b and 6 c are flow charts describing steps in manufacturingintegrated circuits; and

[0064]FIG. 7 is a schematic of a beam writing system employing theinterferometry system.

DETAILED DESCRIPTION OF THE INVENTION

[0065] The present invention relates to apparatus and methods by whichthe refractivity of a gas in at least one measurement path and/or thechange in the optical path length of the measurement path due to the gasmay be quickly measured and used in subsequent downstream orcontemporaneous applications wherein either or both the refractive indexof the gas and the physical length of the measurement path may bechanging. An example of a contemporaneous application is in aninterferometric distance measuring instrument to enhance accuracy bycompensating for the effects of the refractive index of the gas in themeasurement path, especially changes in the optical path length thattake place during the measuring period because of changing environmentalconditions or air turbulence induced in the measurement path by rapidstage slew rates.

[0066] A number of different embodiments of the apparatus of theinvention are shown and described. While they differ in some details,the disclosed embodiments otherwise share many common elements andnaturally fall into two categories depending on the degree of controldemanded of their light sources. As will be seen, the disclosedembodiments within each category also differ in the details of how theirinterferometric optical paths are implemented and/or how certaininformation signals are handled electronically.

[0067] The first group of embodiments to be described comprise twoembodiments and variants thereof. This group is intended forapplications where the stability of the adopted light sources issufficient and the ratio of the wavelengths of the light beams generatedby the adopted light sources is matched to a sequence of known ratiovalues with respective relative precisions sufficient to meet therequired precision imposed on the output data by the final end useapplication.

[0068] The second group of embodiments also comprise two embodiments andvariants thereof and these are particularly suitable for use where it isnecessary to monitor the stability of the light sources and measure theratios of the wavelengths of the light beams generated by the adoptedlight sources to meet performance requirements on accuracy. For bothgroups, apparatus is disclosed for dealing with phase ambiguities andphase and group delays that may arise in analyzing homodyne, heterodyne,and/or superheterodyne signals and methods are disclosed forimplementing the steps of the invention.

[0069]FIGS. 1a and 1 b depict in schematic form one preferred embodimentof the present invention for measuring and monitoring the refractivityof a gas in a measurement path and/or the change in the optical pathlength of the measurement path due to the gas wherein either or both therefractive index of the gas and the physical length of the measurementpath may be changing and where the stability of the adopted lightsources is sufficient and the ratio of the wavelengths of the lightbeams generated by the adopted light sources is matched to a known ratiovalue with a relative precision sufficient to meet the requiredprecision imposed on the output data by the final end use application.While the apparatus has application for a wide range of radiationsources, the following description is taken by way of example withrespect to an optical measuring system.

[0070] Referring to FIG. 1a and in accordance with the preferredapparatus and method of the first preferred embodiment of the presentinvention, a light beam 7 emitted from source 1 passes through amodulator 3 becoming light beam 9. Modulator 3 is excited by a driver 5.Source 1 is preferably a laser or like source of coherent radiation,preferably polarized, and having a wavelength λ₁. Modulator 3 may forexample be an acousto-optical device or a combination of acousto-opticaldevices with additional optics for selectively modulating polarizationcomponents of beam 7. Modulator 3 preferably shifts the oscillationfrequency of one linearly polarized component of beam 7 an amount f₁with respect to an orthogonally linearly polarized component, thedirections of polarizations of the components denoted herein as x and y.In the following description of the first preferred embodiment, it willbe assumed that the x polarization component of beam 9 has anoscillation frequency shifted an amount f₁ with respect to the ypolarization component of beam 9 without departing from the spirit orscope of the present invention. The oscillation frequency f₁ isdetermined by the driver 5.

[0071] In a next step, a light beam 8 emitted from a source 2 passesthrough a modulator 4 becoming light beam 10. Modulator 4 is excited bya driver 6, similar to modulator 3 and driver 5, respectively. Source 2,similar to source 1, is preferably a laser or like source of polarized,coherent radiation, but preferably at a different wavelength, λ₂,wherein the ratio of the wavelengths (λ₁/λ₂) has a known approximateratio value l₁/l₂, i.e.

(λ₁/λ₂)≅(l₁/l₂),  (1)

[0072] where l₁ and l₂ may assume integer and non-integer values, andthe ratio of the wavelengths (λ₁/λ₂) is the same as the ratio valuel₁/l₂ to a relative precision of an order of magnitude or more less thanthe dispersion of the refractive index of the gas, (n₂−n₁), times therelative precision ε desired for the measurement of the refractivity ofthe gas or of the change in the optical path length of the measurementleg due to the gas. The x polarized component of beam 10 has anoscillation frequency shifted an amount f₂ with respect to the ypolarized component of beam 10. The oscillation frequency f₂ isdetermined by the driver 6. In addition, the directions of the frequencyshifts of the x components of beams 9 and 10 are the same.

[0073] It will be appreciated by those skilled in the art that beams 7and 8 may be provided alternatively by a single laser source emittingmore than one wavelength, by a single laser source combined with opticalfrequency doubling means to achieve frequency doubling, tripling,quadrupling, etc., two laser sources of differing wavelengths combinedwith sum-frequency generation or difference-frequency generation, or anyequivalent source configuration capable of generating light beams of twoor more wavelengths.

[0074] A laser source, for example, can be a gas laser, e.g. a HeNe,stabilized in any of a variety of conventional techniques known to thoseskilled in the art, see for example, T. Baer et al., “FrequencyStabilization of a 0.633 μm He—Ne-longitudinal Zeeman Laser,” AppliedOptics, 19, 3173-3177 (1980); Burgwald et al., U.S. Pat. No. 3,889,207,issued Jun. 10, 1975; and Sandstrom et al., U.S. Pat. No. 3,662,279,issued May 9, 1972. Alternatively, the laser can be a diode laserfrequency stabilized in one of a variety of conventional techniquesknown to those skilled in the art, see for example, T. Okoshi and K.Kikuchi, “Frequency Stabilization of Semiconductor Lasers forHeterodyne-type Optical Communication Systems,” Electronic Letters, 16,179-181 (1980) and S. Yamaqguchi and M. Suzuki, “SimultaneousStabilization of the Frequency and Power of an AlGaAs SemiconductorLaser by Use of the Optogalvanic Effect of Krypton,” IEEE J. QuantumElectronics, QE-19, 1514-1519 (1983).

[0075] It will also be appreciated by those skilled in the art that thetwo optical frequencies of beam 9 and of beam 10 may be produced by anyof a variety of frequency modulation apparatus and/or lasers: (1) use ofa Zeeman split laser, see for example, Bagley et al., U.S. Pat. No.3,458,259, issued Jul. 29, 1969; G. Bouwhuis, “Interferometrie MitGaslasers,” Ned. T. Natuurk, 34, 225-232 (Aug. 1968); Bagley et al.,U.S. Pat. No. 3,656,853, issued Apr. 18, 1972; and H. Matsumoto, “Recentinterferometric measurements using stabilized lasers,” PrecisionEngineering, 6(2), 87-94 (1984); (2) use of a pair of acousto-opticalBragg cells, see for example, Y. Ohtsuka and K. Itoh, “Two-frequencyLaser Interferometer for Small Displacement Measurements in a LowFrequency Range,” Applied Optics, 18(2), 219-224 (1979); N. Massie etal., “Measuring Laser Flow Fields With a 64-Channel HeterodyneInterferometer,” Applied Optics, 22(14), 2141-2151 (1983); Y. Ohtsukaand M. Tsubokawa, “Dynamic Two-frequency Interferometry for SmallDisplacement Measurements,” Optics and Laser Technology, 16, 25-29(1984); H. Matsumoto, ibid.; P. Dirksen, et al., U.S. Pat. No.5,485,272, issued Jan. 16, 1996; N. A. Riza and M. M. K. Howlader,“Acousto-optic system for the generation and control of tunablelow-frequency signals,” Opt. Eng., 35(4), 920-925 (1996); (3) use of asingle acousto-optic Bragg cell, see for example, G. E. Sommargren,commonly owned U.S. Pat. No. 4,684,828, issued Aug. 4, 1987; G. E.Sommargren, commonly owned U.S. Pat. No. 4,687,958, issued Aug. 18,1987; P. Dirksen, et al., ibid.; (4) use of two longitudinal modes of arandomly polarized HeNe laser, see for example, J. B. Ferguson and R. H.Morris, “Single Mode Collapse in 6328 A HeNe Lasers,” Applied Optics,17(18), 2924-2929 (1978); or (5) use of birefringent elements or thelike internal to the laser, see for example, V. Evtuhov and A. E.Siegman, “A ‘Twisted-Mode’ Technique for Obtaining Axially UniformEnergy Density in a Laser Cavity,” Applied Optics, 4(1), 142-143 (1965).

[0076] The specific device used for the sources of beams 7 and 8 willdetermine the diameter and divergence of beams 7 and 8, respectively.For some sources, e.g., a diode laser, it will likely be necessary touse conventional beam shaping optics, e.g., a conventional microscopeobjective, to provide beams 7 and 8 with a suitable diameter anddivergence for the elements that follow. When the source is a HeNelaser, for example, beam shaping optics may not be required.

[0077] It will be further appreciated by those skilled in the art thatboth the x and y polarization components of beam 9 and/or of beam 10 maybe frequency shifted without departing from the scope and spirit of theinvention, f₁ remaining the difference in frequencies of the x and ypolarization components of beam 9 and f₂ remaining the difference infrequencies of the x and y polarization components of beam 10. Improvedisolation of an interferometer and a laser source is generally possibleby frequency shifting both x and y polarization components of a beam,the degree of improved isolation depending on the means used forgenerating the frequency shifts.

[0078] In a next step, beam 9 is reflected by mirror 253A and then aportion of beam 9 is subsequently reflected by beamsplitter 253B,preferably a non-polarizing type, to become a component of beam 213, theλ₁ component. A portion of beam 10 is transmitted by beamsplitter 253Bto become a second component of beam 213, the λ₂ component, wherein theλ₂ component is preferably parallel and coextensive with the λ₁component. In a further step, beam 213 propagates to an interferometer260, comprised of optical means for a introducing a phase shift φ₁between the polarization components x and y of the λ₁ component of beam213 and a phase shift φ₂ between the polarization components x and y ofthe λ₂ component of beam 213. The magnitude of phase shifts φ₁ and φ₂are related to round-trip physical length L of measurement path 298according to the formulae

φ_(j) =Lpk _(j) n _(j)+ζ_(j), j=1 and 2,  (2)

[0079] where p is the number of passes through the respective referenceand measurement legs for a multiple pass interferometer, and n_(j) arethe refractive indices of gas in measurement path 298 corresponding towavenumber k_(j)=(2π)/λ_(j). The phase offsets ζ_(j) comprise allcontributions to the phase shifts φ_(j) that are not related to themeasurement path 298 or reference paths.

[0080] As shown in FIG. 1a, interferometer 260 is comprised of areference retroreflector 295, object retroreflector 296, quarter-wavephase retardation plates 221 and 222, and a polarizing beam splitter223. This configuration is known in the art as a polarized Michelsoninterferometer, and is shown as a simple illustration with p=1.

[0081] Eqs. (2) are valid for the case where the paths for onewavelength and the paths for the second wavelength are substantiallycoextensive, a case chosen to illustrate in the simplest manner thefunction of the invention in the first embodiment. To those skilled inthe art, the generalization to the case where the respective paths forthe two different wavelengths are not substantially coextensive is astraight forward procedure.

[0082] Cyclic errors that produce nonlinearities in distance measuringinterferometry (cf. the cited articles by Bobroff) have been omitted inEqs. (2). Techniques known to those skilled in the art can be used toeither reduce the cyclic errors to negligible levels or compensate forthe presence of cyclic errors, techniques such as using separated beamsin the interferometer and/or separated beams in the delivery system forlight beams from each light beam source to the interferometer [M.Tanaka, T. Yamagami, and K. Nakayama, “Linear Interpolation of PeriodicError in a Heterodyne Laser Interferometer at Subnanometer Levels,” IEEETrans. Instrum. and Meas., 38(2), 552-554, 1989] and light beam sourceswith reduced polarization and/or frequency mixing.

[0083] After passing through interferometer 260, beam 213 becomes aphase-shifted beam 215, which passes through a polarizer 244 preferablyorientated so as to mix polarization components x and y of beam 215. Aconventional dichroic beam splitter 280 preferably separates thoseportions of beam 215 corresponding to wavelengths λ₁ and λ₂ into beams217 and 218, respectively.

[0084] In a next step as shown in FIG. 1a, phase-shifted beams 217 and218 impinge upon photodetectors 85 and 86, respectively, resulting intwo electrical interference signals, heterodyne signals s₁ and s₂,respectively, preferably by photoelectric detection. The signal s₁corresponds to wavelength λ₁ and signal s₂ corresponds to the wavelengthλ₂. The signals s_(j) have the form

s _(j) =A _(j)cos[α_(j)(t)], j=1 and 2,  (3)

[0085] where the time-dependent arguments α_(j)(t) are given by

α_(j)(t)=2πf _(j) t+φ _(j), j=1 and 2.  (4)

[0086] Heterodyne signals s₁ and s₂ are transmitted to electronicprocessor 109 for analysis as electronic signals 103 and 104,respectively, in either digital or analog format, preferably in digitalformat.

[0087] A preferred method for electronically processing the heterodynesignals s₁ and s₂ is presented herewithin for the case when l₁ and/or l₂are not low order integers. For the case when l₁ and l₂ are both loworder integers and the ratio of the wavelengths matched to the ratio(l₁/l₂) with a relative precision sufficient to meet the requiredprecision imposed on the output data by the end use application, thepreferred procedure for electronically processing the heterodyne signalss₁ and s₂ is the same as the one subsequently set down for the secondvariant of the first preferred embodiment of the present invention.

[0088] Referring now to FIG. 1b, electronic processor 109 furthercomprises electronic processors 1094A and 1094B to determine the phasesφ₁ and φ₂, respectively, by either digital or analog signal processes,preferably digital processes, using time-based phase detection such as adigital Hilbert transform phase detector [see section 4.1.1 of“Phase-locked loops: theory, design, and applications” 2nd ed.McGraw-Hill (New York) 1993, by R. E. Best] or the like and the phase ofdrivers 5 and 6.

[0089] The phases of drivers 5 and 6 are transmitted by electricalsignals, reference signals 101 and 102, respectively, in either digitalor analog format, preferably in digital format, to electronic processor109. Reference signals, alternatives to reference signals 101 and 102,may also be generated by an optical pick off means and detectors (notshown in figures) by splitting off portions of beams 9 and 10 with beamsplitters, preferably non-polarizing beam splitters, mixing the portionof the beam 9 and the portion of the beam 10 that are split off, anddetecting the mixed portions to produce alternative heterodyne referencesignals.

[0090] Referring again to FIG. 1b, phase φ₁ and phase φ₂ are nextmultiplied by l₁/p and l₂/p, respectively, in electronic processors1095A and 1095B, respectively, preferably by digital processing,resulting in phases (l₁/p)φ₁ and (l₂/p)φ₂, respectively. The phases(l₁/p)φ₁ and (l₂/p)φ₂ are next added together in electronic processor1096A and subtracted one from the other in electronic processor 1097A,preferably by digital processes, to create the phases θ and Φ,respectively. Formally, $\begin{matrix}{{\vartheta = \left( {{\frac{l_{1}}{p}\phi_{1}} + {\frac{l_{2}}{p}\phi_{2}}} \right)},} & (5) \\{\Phi = {\left( {{\frac{l_{1}}{p}\phi_{1}} - {\frac{l_{2}}{p}\phi_{2}}} \right).}} & (6)\end{matrix}$

[0091] The phases φ₁, θ, and Φ are transmitted to computer 110 as signal105, in either digital or analog format, preferably in digital format.

[0092] For a measuring path comprised of a vacuum, phase Φ shouldsubstantially be a constant independent of Doppler shifts due to amotion of retroreflector 296. This may not be the case in practice dueto differences in the group delay experienced by the electrical signalss₁ and s₂. Group delay, often called envelope delay, describes the delayof a packet of frequencies, and the group delay at a particularfrequency is defined as the negative of the slope of the phase curve atthe particular frequency [see H. J. Blinchikoff and A. I. Zverev,Filtering in the Time and Frequency Domains, Section 2.6, 1976 (Wiley,New York)]. If phase Φ is not a constant for a measuring path comprisedof a vacuum, techniques known to those skilled in the art can be used tocompensate for departures of phase Φ from a constant (cf. Blinchikoffand Zveriv, ibid.). It is important to note that the group delay effectsin Φ can not only be detected but can also be determined by measuring Φas a function of different translational velocities of retroreflector296 produced by translator 267 for a measuring path comprising a vacuum.It is also important to note that the group delay effects in Φ can besignificantly reduced by performing analog-to-digital conversion ofsignals s₁ and s₂ as close as practical to the photoelectric detectorsin detectors 85 and 86, respectively, followed by digital signalprocessing as opposed to transmitting the signals s₁ and s₂ as analogsignals for subsequent analog signal processing and/or analog-to-digitalconversion downstream. The compensation for a particular group delay cangenerally be introduced before or after, or in part before and in partafter, the processing elements producing the particular group delay.

[0093] The refractivity of the gas, (n₁−1), can be calculated using theformula $\begin{matrix}{{{n_{1} - 1} = {\frac{\Gamma}{\chi^{L}\left\lbrack {1 - \left( {K/\chi} \right)^{2}} \right\rbrack}\left\{ {\left\lbrack {{\vartheta \left( {K/\chi} \right)} - \Phi} \right\rbrack - Q} \right\}}},} & (7)\end{matrix}$

[0094] where

χ=(l ₁ k ₁ +l ₂ k ₂)/2,  (8)

K=(l ₁ k ₁ −l ₂ k ₂)/2  (9)

[0095] and $\begin{matrix}{{\Gamma = \frac{n_{1} - 1}{n_{2} - n_{1}}},} & (10)\end{matrix}$

[0096] the quantity Γ being the reciprocal dispersive power of the gaswhich is substantially independent of environmental conditions andturbulence in the gas. The offset term Q is defined as

Q=ξ(K/χ)−Z  (11)

[0097] where $\begin{matrix}{{\xi = \left( {{\frac{l_{1}}{p}\zeta_{1}} + {\frac{l_{2}}{p}\zeta_{2}}} \right)},} & (12) \\{Z = {\left( {{\frac{l_{1}}{p}\zeta_{1}} - {\frac{l_{2}}{p}\zeta_{2}}} \right).}} & (13)\end{matrix}$

[0098] Values of Γ may be computed from knowledge of the gas compositionand from knowledge of the wavelength dependent refractivities of the gasconstituents. For the example of λ₁=0.63 μm, λ₂=0.32 μm, and a standardatmosphere, Γ≅24.

[0099] In addition, Eq. (7) is valid for the case where the combinedpaths for optical beams at one wavelength are substantially coextensivewith the combined paths for optical beams at a second wavelength, apreferred configuration that also serves to illustrate in the simplestmanner the function of the invention. To those skilled in the art, thegeneralization to the case where combined paths for optical beams at onewavelength are not substantially coextensive with the combined paths foroptical beams at a second wavelength is a straight forward procedure.

[0100] For those applications related to distance measuringinterferometry, the heterodyne phase φ₁ and phases θ and Φ may be usedto determine a physical distance L, independent of the effects of therefractive index of the gas in the measuring path of a distancemeasuring interferometer, using the formula $\begin{matrix}{L = {\frac{1}{\left( {\chi + K} \right)}{\left\{ {{\frac{l_{1}}{p}\left( {\phi_{1} - \zeta_{1}} \right)} - {\frac{\Gamma}{\left\lbrack {1 - \left( {K/\chi} \right)} \right\rbrack}\left\lbrack {{\left( {K/\chi} \right)\vartheta} - \Phi - Q} \right\rbrack}} \right\}.}}} & (14)\end{matrix}$

[0101] The ratio of the wavelengths can be expressed in terms of (K/χ)from Eqs. (8) and (9) with the result $\begin{matrix}{\frac{\lambda_{1}}{\lambda_{2}} = {{\left( \frac{l_{1}}{l_{2}} \right)\left\lbrack \frac{1 - \left( {K/\chi} \right)}{1 + \left( {K/\chi} \right)} \right\rbrack}.}} & (15)\end{matrix}$

[0102] When operating under the condition $\begin{matrix}{{{{K/\chi}}\frac{\left( {n_{2} - n_{1}} \right)}{\left( {n_{2} + n_{1}} \right)}},} & (16)\end{matrix}$

[0103] the ratio of the phases Φ and θ has the approximate value$\begin{matrix}{\left( {\Phi/\vartheta} \right) \cong {- {\frac{\left( {n_{2} - n_{1}} \right)}{\left( {n_{2} + n_{1}} \right)}.}}} & (17)\end{matrix}$

[0104] Therefore, for the case of the first preferred embodiment wherethe ratio of the wavelengths (λ₁/λ₂) has a known approximate ratio valuel₁/l₂, cf. Eq. (1), where l₁ and l₂ may assume integer and non-integervalues, and the ratio of the wavelengths (λ₁/λ₂) is the same as theratio value l₁/l₂ to a relative precision of an order of magnitude ormore less than the dispersion of the refractive index of the gas,(n₂−n₁), times the relative precision ε desired for the measurement ofthe refractivity of the gas or of the change in the optical path lengthof the measurement leg due to the gas, expressed formally by theinequality $\begin{matrix}{{{{\frac{\lambda_{1}}{\lambda_{2}} - \frac{l_{1}}{l_{2}}}}{\left( \frac{l_{1}}{l_{2}} \right)\left( {n_{2} - n_{1}} \right)ɛ}},} & (18)\end{matrix}$

[0105] Eqs. (7) and (14) reduce to more simple forms of $\begin{matrix}{{{n_{1} - 1} = {{- \frac{\Gamma}{\chi^{L}}}\left( {\Phi + Q} \right)}},} & (19)\end{matrix}$

$\begin{matrix}{{L = {\frac{1}{\chi}\left\lbrack {{\frac{l_{1}}{p}\left( {\phi_{1} - \zeta_{1}} \right)} + {\Gamma \left( {\Phi + Q} \right)}} \right\rbrack}},} & (20)\end{matrix}$

[0106] respectively. It will also be obvious to someone skilled in theart to perform similar calculations for L with respect to n₂,

(n ₂−1)=(n ₁−1)(1+1/Γ),  (21)

[0107] in place of or in addition to n₁.

[0108] In a next step, electronic processing means 109 transmits to thecomputer 110 φ₁ and Φ as electronic signal 105 in either digital oranalog format, preferably a digital format, for the computation of(n₁−1) and/or L. The resolution of phase redundancy in (1/l₁)Φ isrequired in the computation of either (n₁−1) or changes in L due to thegas using either Eqs. (19) or (20), respectively. In addition, theresolution of phase redundancy in φ₁ is required in the computation of Lusing Eq., and the resolution of the phase redundancy in φ₁ is requiredin the computation of changes L using Eq. (20) if χ is variable in time.

[0109] The equivalent wavelength comprising (1/l₁)Φ is significantlylarger than either of the wavelengths λ₁ and λ₂ and as a consequence,produces a significant simplification in a procedure implemented forresolution of phase redundancy in (1/l₁)Φ. The equivalent wavelengthλ_((1/l) ₁ _()Φ) for (1/l₁)Φ is $\begin{matrix}{\lambda_{{({1/l_{1}})}\Phi} = {\frac{\lambda_{1}}{\left( {n_{2} - n_{1}} \right)}.}} & (22)\end{matrix}$

[0110] For the example of λ₁=0.63 μm, λ₂ =0.32 μm, and (n ₂−n₁)≅1×10⁻⁵for a standard atmosphere, the equivalent wavelength given by Eq. (22)is

λ_((l/l) ₁ _()Φ)≅63 mm.  (23)

[0111] Any one of several procedures may be easily employed to resolvethe phase redundancy in (1/l₁)Φ, given the equivalent wavelength asexpressed by Eq. (22). For those applications where changes in themeasurement path can be measured interferometrically, a feature forexample of an application based on a distance measuring interferometeremployed for measuring changes in the measurement path, the movableretroreflector 296 of interferometer 260 can be scanned by translator267 in a controlled manner over a given length and the concomitantchange in (1/l₁)Φ recorded. From the recorded change in (1/l₁)Φ and thelength scanned, as recorded by the change in φ₁, the equivalentwavelength λ_((1/l) ₁ _()Φ) can be calculated. With the computed valuefor the equivalent wavelength λ_((1/l) ₁ _()Φ), the phase redundancy in(1/l₁)Φ can be easily resolved in view of the relatively large value forthe equivalent wavelength λ_((1/l) ₁ _()Φ).

[0112] For those applications where the determination of therefractivity and/or or the change in the optical path length due to thegas in a measurement leg is made and retroreflector 296 does not have ascanning capability such as considered in the preceding paragraph, otherprocedures are available for the resolution of the phase redundancy of(1/l₁)Φ. One procedure which may be employed to resolve the phaseredundancy in (1/l₁)Φ is based on the use of a retroreflector or seriesof retroreflectors inserted at a series of positions in measurement path298, the retroreflector at each position of the series of positionsperforming the same function as retroreflector 296, wherein theround-trip physical length L for the measurement leg for each positionof the series of positions of the inserted retroreflectors form ageometric progression. The smallest or first round-trip physical lengthin the series of positions will be approximately λ₁/[4(n₂−n₁)] dividedby the relative precision that the initial value of (1/l₁)Φ is known.The round-trip physical length of the second position of the series ofpositions will be approximately the round-trip physical length of thefirst position of the series of positions divided by the relativeprecision that Φ is measured using the first position of the series ofpositions. This is a geometric progression procedure, the resultingsequence of round-trip physical lengths forming a geometric progression,which is continued until the round-trip physical length of theretroreflector 296 used to measure the refractivity or the change inoptical path length due to the refractivity of the gas would be exceededif the number of positions in the series of positions were incrementedby one.

[0113] A third procedure is based upon the use of a source (not shown inFIGS. 1a and 1 b) of a series of known wavelengths and measuring Φ forthese wavelengths. The number of known wavelengths required for theresolution of the phase redundancy is generally comprised of a small setbecause of the relatively large value for λ_((1/l) ₁ _()Φ) as given byEq. (22).

[0114] Another procedure to resolve the phase redundancy in (1/l₁)Φwould be to observe the changes in (1/l₁)Φ as the measuring path 98 ischanged from being filled with a gas to an evacuated state (the vacuumchamber and pump and requisite gas handling system are not shown inFIGS. 1a and 1 b) to resolve the phase redundancy in (1/l₁)Φ. Theproblems normally encountered in measuring absolute values forrefractivity and changes in the optical path length due to therefractivity of the gas based in part on changing the gas pressure froma non-zero value to a vacuum are not present in the first preferredembodiment because of the relative large equivalent wavelength of(1/l₁)Φ as expressed by Eq. (22).

[0115] The resolution of the phase redundancy in φ₁, if required,presents a problem similar to the one as subsequently described withrespect to the required resolution of phase redundancy in θ in thefourth embodiment and variants thereof of the present invention. As aconsequence, the procedure described for the resolution of phaseredundancy in θ with respect to the fourth embodiment and variantsthereof can be adapted for use in the resolution of the phase redundancyin φ₁.

[0116] The offset terms involving ζ₁ or/and Q that are present in Eqs.(19) and (20) and defined in Eqs. (2) and (11) are terms that requiresome combination of determination and/or monitoring depending on whetherχ is variable in time, whether the refractivity or/and the length L areto be determined, respectively, or whether changes in refractivityor/and the length L are to be determined, respectively. Thedetermination and/or monitoring of ζ₁ or/and Q as required presents aproblem similar to the one as subsequently described with respect to thedetermination and/or monitoring of ζ₃ and/or Q in the second and fourthembodiments and variants thereof of the present invention. As aconsequence, the procedures described for the determination and/ormonitoring of ζ₃ and/or Q with respect to the second and fourthembodiments and variants thereof can be adapted for use in the firstembodiment for the determination and/or monitoring of ζ₁ and/or Q asrequired.

[0117] A first variant of the first preferred embodiment is disclosedwherein the description of the apparatus of the first variant of thefirst embodiment is the same as that given for the apparatus of thefirst embodiment except with regard to the detection of beams 217 and218 of the first embodiment shown in FIG. 1a. In the first variant ofthe first embodiment, a first portion of beam 217 is detected by adetector (not shown in the figures) creating a signal proportional tos₁, as₁ where a is a constant, and beam 218 and a second portion of beam217 are detected by a second single detector (not shown in the figures)creating signal S_(b1+2)=bs₁+s₂ where b is a constant. Heterodynesignals as₁ and S_(b1+2) are transmitted as electronic signals 1103 and1104, respectively, in either digital or analog format, preferably indigital format, to electronic processor 109A shown in diagrammatic formin FIG. 1c for analysis.

[0118] Referring now to FIG. 1c, electronic processor 109A preferablycomprises alphameric numbered elements wherein the numeric component ofthe alphameric numbers indicate the function of an element, the samenumeric component/function association as described for the electronicprocessing elements of the first embodiment depicted in FIG. 1b. Thedescription of the steps performed by electronic processor 109A inprocessing heterodyne signals bs₁ and s₂ comprising S_(b1+2) for phasesθ and Φ is the same as corresponding portions, according to the numericcomponent of the alphameric numbers of elements, of the description ofsteps performed by electronic processor 109 in processing heterodynesignals s₁ and s₂ of the first embodiment. The description of the stepsin processing heterodyne signal as₁ by electronic processor 109A forphase φ₁ is the same as corresponding portions, according to the numericcomponent of the alphameric numbers of elements, of the description ofsteps in the processing of the heterodyne signal s₁ of the firstembodiment by electronic processor 109.

[0119] The phases φ₁, θ, and Φ created by electronic processor 109Aformally have the same properties as φ₁, θ, and Φ, respectively, createdby electronic processor 109 of the second embodiment.

[0120] The feature of the first variant of the first embodiment whichcan be a significant feature is the detection the optical beams creatingheterodyne signals bs₁ and s₂ by a single detector. It will be apparentto those skilled in the art that the single detector feature of thefirst variant of the first embodiment can be important in reducing oreliminating the effects of differences in certain group delays possiblein the first embodiment. The remaining description of the first variantof the first embodiment is the same as corresponding portions of thedescription given for the first embodiment.

[0121] A second variant of the first preferred embodiment is disclosedwherein the description of the apparatus of the second variant of thefirst embodiment is the same as that given for the apparatus of thefirst embodiment except with regard to the frequencies f₁ and f₂ ofdrivers 5 and 6, respectively, shown in FIG. 1a. In the second variantof the first embodiment, the frequencies of the two drivers 5 and 6 arethe same, i.e. f₁=f₂. This feature of the second variant of the firstembodiment eliminates the effects of differences in group delays in thefirst embodiment resulting from f₁≠f₂. The remaining description of thesecond variant of the first embodiment is the same as the correspondingportions of the description given for the first preferred embodiment.

[0122] Reference is now made to FIGS. 1a and 1 d which taken togetherdepict in diagrammatic form a third variant of the first preferredembodiment of the present invention for measuring and monitoring therefractivity of a gas in a measurement path and/or the change in theoptical path length of the measurement path due to the gas whereineither or both the refractive index of the gas and the physical lengthof the measurement path may be changing and where the stability of theadopted light sources is sufficient and the wavelengths of the lightbeams generated by the adopted light sources are harmonically related toa relative precision sufficient to meet the required precision imposedon the output data by the final end use application. The conditionwherein the wavelengths are approximately harmonically relatedcorresponds to the special case of the first embodiment in which theratio (l₁/l₂) is expressible as the ratio of low order non-zero integers(p₁/p₂), i.e. $\begin{matrix}{{l_{1} = p_{1}},{l_{2} = p_{2}},{\left( \frac{l_{1}}{l_{2}} \right) = \left( \frac{p_{1}}{p_{2}} \right)},p_{1},{p_{2} = 1},2,\ldots \quad,{p_{1} \neq {p_{2}.}}} & (24)\end{matrix}$

[0123] The description of the sources of light beams 9 and 10 and oflight beams 9 and 10 for the second variant of the first embodiment isthe same as that for description of the sources of light beams 9 and 10and of light beams 9 and 10 given for the first embodiment with theadditional requirement that the wavelengths be harmonically related to arelative precision sufficient to meet the required precision imposed onthe output data by the final end use application. The description of theapparatus for the second variant of the first embodiment depicted inFIG. 1a is the same as corresponding portions of the description givenfor the first embodiment.

[0124] Referring now to FIG. 1d, electronic processing means 109Bpreferably comprises means 1092A and 1092B for electronicallymultiplying time-dependent arguments α₁(t) and α₂(t), respectively, ofheterodyne signals s₁ and s₂, respectively, by coefficients p₁ and p₂,respectively, so as to create two modified heterodyne signals {tildeover (s)}₁ and {tilde over (s)}₂ having the form

{tilde over (s)} _(j) =Ã _(j)cos[p _(j)α_(j)(t)], j=1 and 2.  (25)

[0125] The multiplication may be achieved by any one of the conventionalfrequency multiplying techniques commonly known in the art, such assignal squaring followed by electronic filtering.

[0126] Referring again to FIG. 1d, electronic processing means 109Bpreferably comprises means 1095E for electronically multiplyingtogether, either as an analog or digital process, preferably a digitalprocess, modified heterodyne signals {tilde over (s)}1 and {tilde over(s)}₂ to create a superheterodyne signalS_({tilde over (1)}×{tilde over (2)}) having the mathematical form

S _({tilde over (1)}×{tilde over (2)}) ={tilde over (s)} ₁ {tilde over(s)} ₂.  (26)

[0127] The superheterodyne signal S_({tilde over (1)}×{tilde over (2)})is comprised of two sidebands with a suppressed carrier and may berewritten as

S _({tilde over (1)}×{tilde over (2)}) =S_({tilde over (1)}×{tilde over (2)}) ⁻  (27)

[0128] where

S _({tilde over (1)}×{tilde over (2)}) ⁺=½Ã ₁ Ã ₂cos(2π{tilde over(v)}t+{tilde over (θ)}),  (28)

S _({tilde over (1)}×{tilde over (2)}) ⁻=½Ã ₁ Ã ₂cos(2π{tilde over(F)}t+{tilde over (Φ)}),  (29)

{tilde over (ν)}=(p ₁ f ₁ +p ₂ f ₂),  (30)

{tilde over (θ)}=(p ₁φ₁ +p ₂φ₂),  (31)

{tilde over (F)}=(p ₁ f ₁ −p ₂ f ₂),  (32)

{tilde over (Φ)}=(p ₁φ₁ −p ₂φ₂).  (33)

[0129] Superheterodyne signal S_({tilde over (1)}×{tilde over (2)})therefore comprised of two sidebands,S_({tilde over (1)}×{tilde over (2)}) ⁺ andS_({tilde over (1)}×{tilde over (2)}) ⁻, of equal amplitude, onesideband with frequency {tilde over (ν)} and phase {tilde over (θ)} anda second sideband with frequency {tilde over (F)} and phase {tilde over(Φ)}.

[0130] Referring once again to FIG. 1d, electronic processor 109Bpreferably comprises processor 1093A to separate the two sidebandsignals S_({tilde over (1)}×{tilde over (2)}) ⁺ andS_({tilde over (1)}×{tilde over (2)}) ⁻, using filtering or any of thelike techniques for separating two signals that are separated infrequency. The frequency {tilde over (F)} of the lower frequencysideband of the superheterodyne signal can be very much smaller than thefrequency {tilde over (ν)} of the higher frequency sideband of thesuperheterodyne signal as subsequently described in the remainingdiscussion of the third variant of the first embodiment, considerablysimplifying the separating task of processor 1093A. Electronic processor109B further comprises processors 1094F and 1094G to determine thephases {tilde over (θ)} and {tilde over (Φ)} using time-based phasedetection such as a Hilbert transform phase detector (see section 4.1.1of “Phase-locked loops: theory, design, and applications” by R. E. Best,ibid.) or the like and the phases of the drivers 5 and 6.

[0131] The phases of the drivers 5 and 6 are transmitted as electricalsignals in either digital or analog format, preferably a digital format,for use as reference signals 101 and 102, respectively, to electronicprocessor 109B. The reference signal for the determination of phases{tilde over (θ)} and {tilde over (Φ)} by way of phase sensitivedetection is generated by mixing reference signals 101 and 102 and highpass and low pass filtering, respectively. Electronic processor 109Badditionally comprises processor 1094A to determine the phase shift φ₁using time-based phase detection or the like, reference signal 101serving as the reference signal in phase sensitive detection.

[0132] Reference signals, alternatives to reference signals 101 and 102,may also be generated by an optical pick off means and detectors (notshown in figures) by splitting off portions of beams 9 and 10 with beamsplitters, preferably non-polarizing beam splitters, mixing the portionof the beam 9 and the portion of beam 10 that are split off, anddetecting the mixed portions to produce heterodyne reference signals.

[0133] The quantities p{tilde over (θ)}, p{tilde over (Φ)}, pξ, and pZof the third variant of the first embodiment are formally the same as θ,Φ, ξ, and Z, respectively, of the first embodiment with l₁=p₁ and l₂=p₂.Thus, the refractivity (n₁−1) of the gas or changes in L due to the gasin the measuring path can be expressed in terms of other quantitiesobtained in the third variant of the first embodiment by use of theknown relationships cited in this paragraph and by the use of Eqs. (11),(12), (13), (19), and (20) with l₁=p₁ and l₂=p₂ as specified by Eqs.(24).

[0134] A preferred embodiment of the invention having been disclosed inthe description of the third variant of the first embodiment, theunderlying advantages of the third variant of the first embodiment willbe made more clear by the following discussion. It is evident from thecalculation of the refractivity by Eq. (7) or the calculation of theeffect of the refractivity of the gas on the optical path by Eq. (14),that the required accuracies to which the superheterodyne sidebandphases {tilde over (θ)} and {tilde over (Φ)} must be determined arerelated to the values of the wavenumbers K and χ. In that the frequency{tilde over (F)} can be very much smaller than the frequency {tilde over(ν)}, and since it is generally easier to calculate the phase with highresolution of an electronic signal of lower frequency, it is generallymost advantageous to rely on a high-accuracy measurement of thesuperheterodyne sideband phase {tilde over (Φ)}. This is readilyachieved in the inventive apparatus when the wavenumbers K and χ arerelated according to Eq. (16), the calculation of the refractivity byEq. (19) or the calculation of the effect of the refractivity of the gason the optical path by Eq. (20) substantially not involving thesuperheterodyne sideband phase {tilde over (θ)}. Further, the magnitudeof the superheterodyne sideband phase {tilde over (Φ)} is less than themagnitude superheterodyne sideband phase {tilde over (θ)}, less by afactor of approximately (n₂−n₁)/(n₂+n₁) as expressed by Eq. (17). Thisgreatly improves in general the phase detection accuracy of the quantity[θ(K/χ)−Φ] that appears in Eqs. (7) and (14) and in particular formoving objects, such as are commonly encountered in microlithographyequipment, which may be coupled to the interferometric apparatus as atas at 267.

[0135] Eq. (18) also forms the basis for a conclusion that sources 1 and2 need not be phase locked for the third variant of the firstembodiment. Eq. (18) is actually a weak condition when viewed in termsof a phase-locked requirement for sources 1 and 2. Consider for anexample a desired precision of ε≅3×10⁻⁶ for measuring the refractivity(n₁−1) of the gas or for the change in the optical path length of themeasurement leg due to the gas, corresponding to a relative distancemeasuring precision of approximately 1×10⁻⁹ in a distance measuringinterferometer, (n₁−1)≅3×10⁻⁴, and (n₂−n₁)≅1×10⁻⁵. For the example, thecondition expressed by Eq. (18) written in terms of source frequenciesν₁, and ν₂ instead of wavelengths λ₁ and λ₂, respectively, is$\begin{matrix}{{{v_{2} - {\frac{p_{1}}{p_{2}}v_{1}}}}{3 \times 10^{- 11}{v_{2}.}}} & (34)\end{matrix}$

[0136] For source wavelengths in the visible part of the spectrum andfor low order integers for p₁ and p₂, Eq. (34) translates into acondition $\begin{matrix}{{{v_{2} - {\frac{p_{1}}{p_{2}}v_{1}}}}{30\quad {{kHz}.}}} & (35)\end{matrix}$

[0137] The result expressed in Eq. (35) is clearly a significantly lessrestrictive condition on the frequencies of sources 1 and 2 than aphase-locked condition.

[0138] The remaining description of the third variant of the firstembodiment is the same as corresponding portions of the descriptionsgiven for the first embodiment.

[0139] It will be appreciated by those skilled in the art thatalternative data processing may be considered for the third variant ofthe first embodiment without departing from the spirit and scope of thepresent invention. For example, it may prove useful to generate themodified heterodyne signals by electronically dividing time-dependentarguments α₁(t) and α₂(t) of heterodyne signals s₁ and s₂, respectively,by coefficients p₂ and p₁, respectively, so as to create two modifiedheterodyne signals {tilde over (s)}₁′ and {tilde over (s)}₂′ having theforms $\begin{matrix}{{{\overset{\sim}{s}}_{1}^{\prime} = {{\overset{\sim}{A}}_{1}^{\prime}\quad {\cos \left\lbrack {{\alpha_{1}(t)}/p_{2}} \right\rbrack}}},{{\overset{\sim}{s}}_{2}^{\prime} = {{\overset{\sim}{A}}_{2}^{\prime}\quad {{\cos \left\lbrack {{\alpha_{2}(t)}/p_{1}} \right\rbrack}.}}}} & (36)\end{matrix}$

[0140] The dividing may be achieved by any one of the conventionalfrequency dividing techniques commonly known in the art, such as the useof phase-locked loops or generation of a rectangle wave signal whichchanges sign at every other zero crossing of the signal whose argumentis being divided by two. The subsequent description of the variant ofthe first embodiment based on modified heterodyne signals {tilde over(s)}₁′ and {tilde over (s)}₂′ is the same as corresponding portions ofthe description of the third variant of the first embodiment based onmodified heterodyne signals {tilde over (s)}₁ and {tilde over (s)}₂.

[0141] Another alternative data processing that may be considered forthe third variant of the first preferred embodiment without departingfrom the spirit and scope of the present invention is the addition ofthe modified heterodyne signals {tilde over (s)}₁ and {tilde over (s)}₂together, rather than multiplying them as in the third variant of thefirst embodiment, resulting in the expression:

S _(S) ={tilde over (s)} ₁ +{tilde over (s)} ₂.  (37)

[0142] A superheterodyne signal would be obtained from S_(A) byconventional techniques commonly known in the art such as square lawdetection or by signal rectification. (cf. Dändliker et al., ibid., andRedman and Wall, ibid.). Further, another alternative signal toS_({tilde over (1)}×{tilde over (2)}) may be generated by selecting theappropriate term in the binomial expansion of (s₁+s₂)^(p+q) through theuse of phase sensitive detection.

[0143] Reference is now made to FIGS. 2a-2 f which depict indiagrammatic form the second preferred embodiment of the presentinvention for measuring and monitoring the refractivity of a gas in ameasurement path and/or the change in the optical path length of themeasurement path due to the gas wherein either or both the refractiveindex of the gas and the physical length of the measurement path may bechanging and where the stability of the adopted light sources issufficient and the ratio of the wavelengths of the light beams generatedby the adopted light sources is matched to a known ratio value with arelative precision sufficient to meet the required precision imposed onthe output data by the final end use application.

[0144] A preferred method for electronically processing the heterodynesignals s₃ and s₄ is presented herewithin for the case when l₁ and/or l₂are not low order integers. For the case when l₁ and l₂ are both loworder integers and the ratio of the wavelengths matched to the ratio(l₁/l₂) with a relative precision sufficient to meet the requiredprecision imposed on the output data by the end use application, thepreferred procedure for electronically processing the heterodyne signalss₁ and s₂ is the same as the one subsequently set down for the secondvariant of the first preferred embodiment of the present invention.

[0145] The second preferred embodiment of the present invention iscomprised of a set of differential plane mirror interferometers, thefirst embodiment being comprised of a polarized Michelsoninterferometer, wherein the differential plane mirror interferometer iswell suited to requirements of micro-lithographic fabrication ofintegrated circuits. The description of the sources of light beams 9 and10 and of light beams 9 and 10 for the second embodiment is the same asthe description of the sources of light beams 9 and 10 and of lightbeams 9 and 10 given for the first preferred embodiment of the presentinvention.

[0146] As illustrated in FIG. 2a, beam 9 is incident on differentialplane mirror interferometer 69 and beam 10 is reflected by mirror 54 asbeam 12 which is incident on differential plane mirror interferometer70. Differential plane mirror interferometers 69 and 70, beam splitter65, and external mirrors furnished by external mirror system 90 compriseinterferometric means for introducing a phase shift φ₃ between the x andy components of beam 9 and a phase shift φ₄ between the x and ycomponents of beam 12.

[0147] A differential plane mirror interferometer measures the opticalpath changes between two external plane mirrors. In addition, it isinsensitive to thermal and mechanical disturbances that may occur in theinterferometer beam splitting cube and associated optical components.Differential plane mirror interferometer 69 has four exit/return beams17, 25, 117, and 125 as shown in FIG. 2b. Beams 17 and 25 originatingfrom one frequency component of beam 9 comprise one measurement leg andbeams 117 and 125 originating from a second frequency component of beam9 comprise a second measurement leg. Beams for which the first frequencycomponent of beam 9 is the sole progenitor are indicated in FIG. 2b bydashed lines and beams for which the second frequency component of beam9 is the sole progenitor are indicated in FIG. 2b by dotted lines.

[0148] Differential plane mirror interferometer 70 has four exit/returnbeams 18, 26, 118, and 126 as shown in FIG. 2c. Beams 18 and 26originating from one frequency component of beam 12 comprise onemeasurement leg and beams 118 and 126 originating from a secondfrequency component of beam 12 comprise a second measurement leg. Beamsfor which the first frequency component of beam 12 is the soleprogenitor are indicated in FIG. 2c by dashed lines and beams for whichthe second frequency component of beam 12 is the sole progenitor areindicated in FIG. 2c by dotted lines.

[0149] Beams 17, 25, 117, and 125 are incident on beam splitter 65 andare transmitted by coating 66, preferably a dichroic coating, as beamsE17, E25, E117, and E125, respectively. Beams E17, E25, E117, and E125are incident on external mirror system 90, illustrated in FIG. 2d, whichresults in beams 27 and 127 shown in FIG. 2b. Beams 127 and 27 containinformation at wavelength λ₁ about the optical path length through thegas in measuring path of external mirror system 90 and about the opticalpath length through a reference path, respectively. Likewise, beams 18,26, 118, and 126 are incident on beam splitter 65 and reflected bydichroic coating 66 as beams E18, E26, E118, and E126, respectively.Beams E18, E26, E118, and E126 are incident on external mirror system90, illustrated in FIG. 2e, which results in beams 28 and 128 shown inFIG. 2c. Beam 128 contains information at wavelength λ₂ about opticalpath lengths through the gas in the measuring path of external mirrorsystem 90 and beam 28 contains information at wavelength λ₂ aboutoptical path lengths through a reference path.

[0150] In FIG. 2b, beam 27 is reflected by mirror 63B, a portion ofwhich is reflected by beam splitter 63A, preferably a non-polarizingtype, to become one component of beam 29. A portion of beam 127 istransmitted by beam splitter 63A to become a second component of beam29. Beam 29 is a mixed beam, the first and second components of beam 29having the same linear polarizations. Beam 29 exits the differentialplane mirror interferometer 69.

[0151] Referring to FIG. 2c, beam 28 is reflected by mirror 58B, aportion of which is reflected by beam splitter 58A, preferably anon-polarizing beam splitter, to become a first component of beam 30. Aportion of beam 128 is transmitted by beam splitter 58A to become asecond component of beam 30. Beam 30 is a mixed beam, the first andsecond components of beam 30 having the same linear polarizations.

[0152] The magnitude of phase shifts φ₃ and φ₄ are related to thedifference L_(i) between the round-trip physical length of path i ofmeasurement path 98 and of reference paths shown in FIGS. 2a-2 eaccording to the formulae $\begin{matrix}{{\phi_{3} = {{\sum\limits_{i = 1}^{i = p}\quad \phi_{3,i}} = {{\sum\limits_{i = 1}^{i = p}\quad {L_{i}k_{1}n_{1i}}} + \zeta_{3}}}},{\phi_{4} = {{\sum\limits_{i = 1}^{i = p}\quad \phi_{4,i}} = {{\sum\limits_{i = 1}^{i = p}\quad {L_{i}k_{2}n_{2i}}} + \zeta_{4}}}},} & (38)\end{matrix}$

[0153] where n_(ji) are the refractive indices of gas in path i ofmeasurement path 98 corresponding to wavenumber k_(j). The nominal valuefor L_(i) corresponds to twice the spatial separation of mirror surfaces95 and 96 in external mirror system 90 (cf. FIGS. 2d and 2 e). The phaseoffsets ζ_(j) comprise all contributions to the phase shifts φ_(j) thatare not related to the measurement path 98 or reference paths. In FIGS.2a-2 e, differential plane mirror interferometers 69 and 70, beamsplitter 65, and external mirror system 90 are configured so that p=2 soas to illustrate in the simplest manner the function of the apparatus ofthe second preferred embodiment of the present invention.

[0154] Eqs. (38) are valid for the case where the paths for onewavelength and the paths for the second wavelength are substantiallycoextensive, a case chosen to illustrate in the simplest manner thefunction of the invention in the second embodiment. To those skilled inthe art, the generalization to the case where the respective paths forthe two different wavelengths are not substantially coextensive is astraight forward procedure.

[0155] Cyclic errors that produce nonlinearities in distance measuringinterferometry (cf. the cited articles by Bobroff) have been omitted inEqs. (38). The description of techniques known to those skilled in theart for either the reduction of the cyclic errors to negligible levelsor for the compensation for the presence of cyclic errors is the same ascorresponding portions of the description given for the first preferredembodiment.

[0156] In a next step as shown in FIG. 2a, phase-shifted beams 29 and 30impinge upon photodetectors 185 and 186, respectively, resulting in twoelectrical interference signals, heterodyne signals s₃ and s₄,respectively, preferably by photoelectric detection. The signal s₃corresponds to wavelength λ₁ and signal s₄ corresponds to the wavelengthλ₂. The signals s_(j) have the form the same as that expressed by Eq.(3) with j=3 and 4. Heterodyne signals s₃ and s₄ are transmitted toelectronic processor 209 for analysis as electronic signals 203 and 204,respectively, in either digital or analog format, preferably in digitalformat.

[0157] Referring now to FIG. 2f, electronic processor 209 preferably iscomprised of alphameric numbered elements wherein the numeric componentof the alphameric numbers indicate the function of an element, the samenumeric component/function association as described for the electronicprocessing elements of the first embodiment depicted in FIG. 1b. Thedescription of the steps in processing of the heterodyne signals s₃ ands₄ by electronic processor 209 for phases θ and Φ is the same ascorresponding portions, according to the numeric component of thealphameric numbers of elements, of the description of steps in theprocessing of the heterodyne signals s₁ and s₂ of the first embodimentby electronic processor 109.

[0158] The phases φ₃, θ, and Φ created by electronic processor 209formally have the same properties as φ₁, θ, and Φ, respectively, createdby electronic processor 109 of the first embodiment. Thus, therefractivity (n₁−1) of the gas or changes in L due to the gas in themeasuring path can be expressed in terms of other quantities obtained inthe second embodiment by use of the known relationships cited in thisparagraph and by the use of Eqs. (19) and (20).

[0159] The resolution of phase redundancy in 93 is required in thecomputation of L using Eq. (20) and the resolution of the phaseredundancy in φ₃ is required in the computation of changes L using Eq.(20) if χ is variable in time. The resolution of the phase redundancy inφ₃, if required, presents a problem similar to the one as subsequentlydescribed with respect to the resolution of phase redundancy in θ in thefourth embodiment of the present invention. As a consequence, theprocedures described for the resolution of phase redundancy in θ withrespect to the fourth embodiment can be adapted for use in theresolution of the phase redundancy in φ₃.

[0160] The offset terms involving ζ₃ and Q that are present in Eqs. (19)and (20) and defined in Eqs. (2) and (11) are terms that may requiredetermination and monitoring depending on whether the refractivity(n₁−1) and/or L are being measured, whether changes in the refractivity(n₁−1) and/or L are being measured, whether ζ₃ and/or Q are variable intime, and/or whether χ is variable in time. One procedure for thedetermination of ζ₃ and Q is based on replacement of mirror 91 of theexternal mirror system 90 with a mirror R91 (not shown in FIGS. 2d and 2e) having a surface R93 corresponding to surface 93 of mirror 91 coatedso as be a reflecting surface for both wavelengths λ₁ and λ₂ andmeasuring the resulting values of φ₃ and Φ. Let the resulting values ofφ₃ and Φ be φ_(3R) and Φ_(R), respectively. The quantities ζ₃ and Q arerelated to φ_(3R) and Φ_(R), respectively, as evident from Eqs. (2) and(19) by the formulae

ζ₃=φ_(3R),  (39)

Q=−Φ_(R).  (40)

[0161] The non-electronic contributions to ζ₃ and Q should besubstantially constant in time because of the significant level ofcompensation that takes place in the differential plane mirrorinterferometers 69 and 70, beam splitter 65, and external mirror system90. The electronic contributions to ζ₃ and Q may be monitored by purelyelectronic means (not shown).

[0162] It will be apparent to someone skilled in the art that as aconsequence of the incorporation of beam splitter 65 in the secondpreferred embodiment, polarizing coating 73 of beam splitter 71 andquarter-wave retardation plate 77 need only meet performancespecifications at λ₁ while polarizing coating 74 of beam splitter 72 andquarter-wave retardation plate 78 need only meet performancespecifications at λ₂. This assignment of critical operations accordingto wavelength as disclosed in the second embodiment is an importantaspect of the present invention, particularly in applications requiringprecision such as the case of micro-lithographic fabrication ofintegrated circuits. However, the assignment of operations according towavelength need not done as disclosed in the second preferredembodiment, e.g. the function of beam splitters 71 and 72 being achievedby a single beam splitter with an appropriately modified polarizingsurface, without departing from the spirit or scope of the presentinvention.

[0163]FIG. 2b depicts in schematic form one embodiment of thedifferential plane mirror interferometer 69 shown in FIG. 2a. Itoperates in the following way: beam 9 is incident on beam splitter 55A,preferably a polarizing beam splitter, with a portion of beam 9 beingtransmitted as beam 13. A second portion of beam 9 is reflected by beamsplitter 55A, subsequently reflected by mirror 55B, and then transmittedby half-wave phase retardation plate 79 as beam 113, the half-wave phaseretardation plate 79 rotating by 90° the plane of polarization of thesecond portion of beam 9 reflected by beam splitter 55A. Beams 13 and113 have the same polarizations but still have different frequencies.The function of beam splitter 55A and mirror 55B is to spatiallyseparate the two frequency components of beam 9 using conventionalpolarization techniques.

[0164] Beams 13 and 113 enter polarizing beam splitter 71, which has apolarizing coating 73, and are transmitted as beams 15 and 115,respectively. Beams 15 and 115 pass through quarter-wave phaseretardation plate 77 and are converted into circularly polarized beams17 and 117, respectively. Beams 17 and 117 are transmitted by beamsplitter 65 with dichroic coating 66, reflected back on themselves bymirrors within external mirror system 90 as illustrated in FIG. 2d, passback through beam splitter 65, and subsequently pass back throughquarter-wave retardation plate 77 and converted into linearly polarizedbeams that are orthogonally polarized to the original incident beams 15and 115. These beams are reflected by polarizing coating 73 to becomebeams 19 and 119, respectively. Beams 19 and 119 are reflected byretroreflector 75 to become beams 21 and 121, respectively. Beams 21 and121 are reflected by polarizing coating 73 to become beams 23 and 123,respectively. Beams 23 and 123 pass through quarter-wave phaseretardation plate 77 and are converted into circularly polarized beams25 and 125, respectively. Beams 25 and 125 are transmitted by beamsplitter 65, reflected back on themselves by mirrors within externalmirror system 90 as illustrated in FIG. 2d, pass back through beamsplitter 65, and subsequently pass back through quarter-wave retardationplate 77 and converted into linearly polarized beams, the linearpolarizations being the same as the linear polarizations of the originalincident beams 15 and 115. These beams are transmitted by polarizingcoating 73 to become beams 27 and 127, respectively.

[0165] Beam 27 is reflected by mirror 63B, and then a portion reflectedby beam splitter 63A, preferably a non-polarizing type, as a firstcomponent of beam 29. Beam 127 is incident on beam splitter 63A with aportion of beam 127 being transmitted as a second component of beam 29,the first and second components of beam 29 having the same linearpolarizations but still having different frequencies. Phase-shifted beam29 is a mixed beam, the first and second components of beam 29 havingthe same linear polarizations.

[0166]FIG. 2c depicts in schematic form one embodiment of differentialplane mirror interferometer 70 shown in FIG. 2a. It operates in thefollowing way: Beam 12 is incident on beam splitter 56A, preferably apolarizing beam splitter, with a portion of beam 12 being transmitted asbeam 14. A second portion of beam 12 is reflected by beam splitter 56A,subsequently reflected by mirror 56B, and then transmitted by half-wavephase retardation plate 80 as beam 114, the half-wave phase retardationplate 80 rotating by 90° the plane of polarization of the second portionof beam 12 reflected by beam splitter 56A. Beams 14 and 114 have thesame polarizations but still have different frequencies. The function,in part, of beam splitter 56A and mirror 56B is to spatially separatethe two frequency components of beam 12 using conventional polarizationtechniques.

[0167] Beams 14 and 114 enter polarizing beam splitter 72, which has apolarizing coating 74, and are transmitted as beams 16 and 116,respectively. Beams 16 and 116 pass through quarter-wave phaseretardation plate 78 and are converted into circularly polarized beams18 and 118, respectively. Beams 18 and 118 are reflected by beamsplitter 65 with dichroic coating 66, reflected back on themselves bymirrors within external mirror system 90 as illustrated in FIG. 2e,reflected by surface 66 of beam splitter 65 a second time, andsubsequently pass back through quarter-wave retardation plate 78 andconverted into linearly polarized beams that are orthogonally polarizedto the original incident beams 16 and 116. These beams are reflected bypolarizing coating 74 to become beams 20 and 120, respectively. Beams 20and 120 are reflected by retroreflector 76 to become beams 22 and 122,respectively. Beams 22 and 122 are reflected by polarizing coating 74 tobecome beams 24 and 124, respectively. Beams 24 and 124 pass throughquarter-wave phase retardation plate 78 and are converted intocircularly polarized beams 26 and 126, respectively. Beams 26 and 126are reflected by surface 66 of beam splitter 65, reflected back onthemselves by mirrors within external mirror system 90 as illustrated inFIG. 2e, reflected by surface 66 of beam splitter 65 a second time, andsubsequently pass back through quarter-wave retardation plate 78 andconverted into linearly polarized beams, the same linear polarizationsas the linear polarizations of the original incident beams 16 and 116.These beams are transmitted by polarizing coating 74 to become beams 28and 128, respectively. Beams 28 and 128 contain information atwavelength λ₂ about the optical path lengths through the gas inmeasurement path 98 wherein the effects of the refractivity of the gasis to be determined and about the optical path lengths through thereference leg, respectively.

[0168] Beam 28 is reflected by mirror 58B, and then a portion reflectedby beam splitter 58A, preferably a non-polarizing type, as a firstcomponent of beam 30. Beam 128 is incident on beam splitter 58A with aportion of beam 128 being transmitted as a second component of beam 30,the first and second components of beam 30 having the same linearpolarizations but still having different frequencies. Phase-shifted beam30 is a mixed beam, the first and second components of beam 30 havingthe same linear polarizations.

[0169] The remaining description of the second preferred embodiment isthe same as corresponding portions of the description given for thefirst preferred embodiment.

[0170] There are three variants of the second embodiment wherein thedescription of each of the three variants of the second embodiment isthe same as corresponding portions of descriptions given for the threevariants of the first preferred embodiment.

[0171] The description of the first preferred embodiment noted that theconfiguration of interferometer illustrated in FIG. 1a is known in theart as a Michelson interferometer. The description of the secondpreferred embodiment noted that the configuration of interferometersillustrated in FIGS. 2a-2 e are known in the art as differential planemirror interferometers. Other forms of the differential plane mirrorinterferometer and forms of other interferometers such as the planemirror interferometer or the angle-compensating interferometer orsimilar device such as is described in an article entitled “Differentialinterferometer arrangements for distance and angle measurements:Principles, advantages and applications” by C. Zanoni, VDI Berichte Nr.749, 93-106 (1989), is preferably incorporated into the apparatus of thefirst embodiment of the present invention as when working with stagescommonly encountered in the micro-lithographic fabrication of integratedcircuits without significantly departing from the spirit and scope ofthe present invention.

[0172] The third and fourth preferred embodiments of the presentinvention and variants thereof, illustrated in FIGS. 3a-3 b and 4 a-4 c,respectively, are embodiments to measure a refractivity of a gas and/orthe change in the optical path length of a measurement path due to thegas when the condition set fourth in Eq. (18) for the first and secondpreferred embodiments and variants thereof is not satisfied, i.e.$\begin{matrix}{{{\frac{\lambda_{1}}{\lambda_{2}} - \frac{l_{1}}{l_{2}}}}{\left( \frac{l_{2}}{l_{1}} \right)\left( {n_{2} - n_{1}} \right){ɛ.}}} & (41)\end{matrix}$

[0173] Under the condition set fourth in Eq. (41), the approximateratio, preferably the ratio (K/χ), must be either known or measured inaccordance with Eqs. (7) and (14) for the third and fourth embodimentsand variants thereof in addition to already described quantities inorder to achieve the required accuracy in the determination of arefractivity of the gas and/or the change in the optical path of themeasurement path due to the gas.

[0174] Each of the first and second preferred embodiments and variantsthereof can be converted from an apparatus and method for measuring arefractivity of the gas and/or the change in the optical path of themeasurement path due to the gas to an apparatus and method for measuringχ and/or the ratio (K/χ). The conversion, as demonstrated in thefollowing descriptions, is accomplished by changing the measurement legsof the first and second embodiments and variants thereof so that themeasuring paths through a gas in respective measurement paths 298 and 98are replaced by a predetermined medium, preferably a vacuum, and therespective measurement legs have a fixed physical length. Accordingly,the third embodiment and variants thereof are each comprised of anunmodified and a modified apparatus and method from the first embodimentor a variant thereof and the fourth embodiment and variants thereof areeach comprised of an unmodified and a modified apparatus and method fromthe second embodiment of a variant thereof, the modified apparatus andmethod being comprised of the respective unmodified apparatus and methodwith a modified measurement leg.

[0175] Reference is now made to FIGS. 3a-3 b that depict in diagrammaticform the third preferred embodiment of the present invention. Thedescription of the source of light beam 9 of the third embodiment is thesame as that for light beam 9 of the first preferred embodiment and thedescription of the source of light beam 10 of the third embodiment isthe same as that for light beam 10 of the first preferred embodimentexcept that the condition on wavelengths λ₁ and λ₂ expressed by Eq. (18)is replaced by the condition set fourth in Eq. (41). Light beam 9 isreflected by mirror 253A and a first portion reflected by beam splitter253B, preferably a non-polarizing type, to become one component of beam213, the λ₁ component. A second portion of beam 9 reflected by mirror253A is transmitted by beam splitter 253B and reflected by mirror 253cto become one component of beam 213b, the λ₁ component. Light beam 10 isincident on beam splitter 253B and a first portion transmitted to becomea second component of beam 213, the λ₂ component. A second portion ofbeam 10 is reflected by beam splitter 253B and reflected by mirror 253 cto become a second component of beam 213 b, the λ₂ component (see FIG.3a). The λ₁ and λ₂ components of beams 213 and 213 b are preferablyparallel and coextensive, respectively.

[0176] Because of the requirement in the third preferred embodiment tomeasure χ and/or the ratio (K/χ), the third preferred embodiment asdescribed in a preceding paragraph is comprised in part of the sameapparatus and method as for the first preferred embodiment and ofadditional means for determination of % and/or the ratio (K/χ). Theadditional means for determination of χ and/or the ratio (K/χ) is thesame as the apparatus and method of the first preferred embodimentexcept for the measurement path 298. Consequently, a number of elementsof the apparatus shown in FIGS. 3a-3 b for determination of χ and/or theratio (K/χ) perform analogous operations as apparatus for determinationof a refractivity of a gas and/or the change in the optical path lengthof a measurement path due to the gas of the first preferred embodiment,apart from the suffix “b” when referring to apparatus for determinationof χ and/or the ratio (K/χ).

[0177] The description of interferometer 260 b is the same as that forinterferometer 260 except with respect to the gas in measurement path298 b and the round-trip physical length of the measurement path 298 b.The measurement leg in the interferometer 260 b of the third preferredembodiment includes measurement path 298 b as illustrated in FIGS. 3a,measurement path 298 b preferably being an evacuated volume of fixedlength (L_(b)/2).

[0178] The differences in the measurement path 298 b and 298 lead tomodifications of Eqs. (2) such that the magnitude of phase shifts φ_(1b)and φ_(2b), counterparts to phase shifts φ₁ and φ₂, respectively, arerelated to the round-trip physical length L_(b) of measurement path 298b and to reference paths as shown in FIG. 3a according to the formulae

φ_(jb) =L _(b) pk _(j)+ζ_(jb), j=1 and 2.  (42)

[0179] In a next step as shown in FIG. 3a, phase-shifted beams 217 b and218 b impinge upon photodetectors 85 b and 86 b, respectively, resultingin two interference signals, heterodyne signals s_(1b) and s_(2b),respectively, preferably by photoelectric detection. The signal s_(1b)corresponds to wavelength λ₁ and signal s_(2b) corresponds to thewavelength λ₂. The signals s_(jb) have the form s _(jb) =A_(jb)cos[α_(jb)(t)], j=1 and 2,  (43)

[0180] where the time-dependent arguments α_(jb)(t) are given by

α_(jb)(t)=2πf _(j) t+φ _(jb), j=1 and 2.  (44)

[0181] Heterodyne signals s_(1b) and s_(2b) are transmitted toelectronic processor 109 b for analysis as electronic signals 103 b and104 b, respectively, in either digital or analog format, preferably indigital format.

[0182] A preferred method for electronically processing the heterodynesignals s_(1b) and s_(2b) is presented herewithin for the case when l₁and/or l₂ are not low order integers. For the case when l₁ and l₂ areboth low order integers and the ratio of the wavelengths matched to theratio (l₁/l₂) with a relative precision sufficient to meet the requiredprecision imposed on the output data by the end use application, thepreferred procedure for electronically processing the heterodyne signalss₁ and s₂ is the same as the one subsequently set down for the thirdvariant of the third preferred embodiment of the present invention.

[0183] Referring now to FIG. 3b, electronic processor 109 b furthercomprises electronic processors 1094Ab and 1094Bb to determine thephases φ_(1b) and φ_(2b), respectively, by either digital or analogsignal processes, preferably digital processes, using time-based phasedetection such as a digital Hilbert transform phase detector [R. E.Best, ibid.] or the like and the phase of drivers 5 and 6.

[0184] Referring again to FIG. 3b, the phase φ_(1b) and the phase φ_(2b)are next multiplied by l₁/p and l₂/p, respectively, in electronicprocessors 1095Ab and 1095Bb, respectively, preferably by digitalprocessing, resulting in phases (l₁/p)φ_(1b) and (l₂/p)φ_(2b),respectively. The phases (l₁/p)φ_(1b) and (l₂/p)φ_(2b) are next addedtogether in electronic processor 1096Ab and subtracted one from theother in electronic processor 1097Ab, preferably by digital processes,to create the phases θ_(1b) and Φ_(1b), respectively. Formally,$\begin{matrix}{{\vartheta_{1b} = \left( {{\frac{l_{1}}{p}\phi_{1b}} + {\frac{l_{2}}{p}\phi_{2b}}} \right)},} & (45) \\{\Phi_{1b} = {\left( {{\frac{l_{1}}{p}\phi_{1b}} - {\frac{l_{2}}{p}\phi_{2b}}} \right).}} & (46)\end{matrix}$

[0185] The phases θ_(1b) and Φ_(1b) are transmitted to computer 110 assignals 105 b, in either digital or analog format, preferably in digitalformat.

[0186] The quantities χ and K are related to phase θ_(b) and phase Φ_(b)according to the formulae

χ=(θ_(b)−ξ_(b))/(2L _(b)),  (47)

K=(Φ_(b) −Z _(b))/(2L _(b)),  (48)

[0187] where $\begin{matrix}{{\xi_{2b} = \left( {{\frac{l_{1}}{p}\zeta_{1b}} + {\frac{l_{2}}{p}\zeta_{2b}}} \right)},} & (49)\end{matrix}$

$\begin{matrix}{Z_{2b} = {\left( {{\frac{l_{1}}{p}\zeta_{1b}} - {\frac{l_{2}}{p}\zeta_{2b}}} \right).}} & (50)\end{matrix}$

[0188] Eqs. (47) and (48) show that within a multiplicative factor[1/(2L_(b))] and phase offset terms ξ_(b) and Z_(b), χ and K are equalto the phase θ_(b) and the phase Φ_(b), respectively.

[0189] The ratio (K/χ) can be expressed by the formula $\begin{matrix}{\frac{K}{\chi} = \frac{\left( {\Phi_{b} - Z_{b}} \right)}{\left( {\vartheta_{b} - \xi_{b}} \right)}} & (51)\end{matrix}$

[0190] using Eqs. (47) and (48). Therefore the ratio (K/χ) is obtainedby substantially dividing Φ_(b) by θ_(b) without the requirement for anaccurate measurement of L to the same precision as required for (K/χ).The phase redundancy of Φ_(b) can be determined as part of the sameprocedure used to remove the phase redundancy of Φ in the unmodifiedapparatus and method of the first preferred embodiment incorporated aspart of the third preferred embodiment.

[0191] The determination of the phase offsets ξ_(b) and Z_(b) is aproblem similar to the one described with respect to the determinationof ξ_(b) and Z_(b) in the fourth preferred embodiment. As a consequence,the procedures described for the determination of ξ_(b) and Z_(b) withrespect to the fourth embodiment can be adapted for determination ofξ_(b) and Z_(b) in the third embodiment.

[0192] The refractivity of the gas and/or the change in the optical pathlength of a measurement path due to the gas is subsequently obtainedusing Eqs. (7) and/or (14), respectively. Because of the non-negligibleeffect of θ in Eqs. (7) and (14), the phase redundancy of θ must also beresolved in addition to the resolution of the phase redundancy of θ_(b).The remainder of the description of the third preferred embodiment isthe same as that given for corresponding aspects of the first preferredembodiment except with respect to the description of the procedure forthe resolution of the phase redundancies of θ and of θ_(b). Theresolution of the phase redundancies in θ and θ_(b) is a problem similarto the one as subsequently described with respect to the requiredresolution of phase redundancy in θ and θ_(b) in the fourth embodimentof the present invention. As a consequence, the procedures described forthe resolution of phase redundancies in θ and θ_(b) with respect to thefourth embodiment can be adapted for use in the resolution of the phaseredundancies in θ and θ_(b) for the third embodiment.

[0193] There are three variants of the third embodiment wherein thedescription of each of the three variants of the third embodiment is thesame as the description given for corresponding portions of the threevariants of the first preferred embodiment.

[0194] Reference is now made to FIGS. 4a-4 c which depict indiagrammatic form the fourth preferred embodiment of the presentinvention. The description of the source of light beams 9 and 9 b of thefourth embodiment is the same as that for light beam 9 of the secondpreferred embodiment and the description of the source of light beams 10and 10 b of the fourth embodiment is the same as that for light beam 10of the second preferred embodiment except that the condition onwavelengths λ₁ and λ₂ expressed by Eq. (18) is replaced by the conditionset fourth in Eq. (41). Light beams 9 and 9 b of the fourth embodimentare derived from a common light beam by beam splitter 153A, preferably anon-polarizing type, and mirror 153B and light beams 10 and 10 b of thefourth embodiment are derived from a common light beam by beam splitter154A, preferably a non-polarizing type, and mirror 154B (see FIG. 4a).

[0195] Because of the requirement in the fourth preferred embodiment tomeasure χ and/or the ratio (K/χ), the fourth preferred embodiment iscomprised in part of the same apparatus and method as for the secondpreferred embodiment and of additional means for determination of χand/or the ratio (K/χ). The additional means for determination of χand/or the ratio (K/χ) is the same as the apparatus and method of thesecond preferred embodiment except for the external mirror system 90 b.Consequently, a number of elements of the apparatus shown in FIGS. 4a-4c for determination of χ and/or the ratio (K/χ) perform analogousoperations as apparatus for determination of a refractivity of a gasand/or the change in the optical path length of a measurement path dueto the gas of the second preferred embodiment, apart from the suffix “b”when referring to apparatus for determination of χ and/or the ratio(K/χ).

[0196] The external mirror system 90 b of the fourth preferredembodiment is shown in FIGS. 4b and 4 c. The description of externalmirror system 90 b is the same as that for external mirror system 90except with respect to the gas in the measurement path 98 and theround-trip physical length of the measurement path 98. The measurementleg in the external mirror system 90 b of the fourth preferredembodiment includes measurement path 98 b as illustrated in FIGS. 4b and4 c, measurement path 98 b preferably being an evacuated volume definedby mirrors 91 b and 92 b and a cylinder 99 b of fixed length (L_(b)/2).Referring to FIGS. 4b and 4 c, surface 95 b is coated so as to reflectwith high efficiency beams E17 b, E25 b, E18 b, and E26 b and totransmit with high efficiency beams E117 b, E125 b, E118 b, and E126 b.Surface 96 b is coated to reflect with high efficiency beams E117 b,E125 b, E118 b, and E126 b.

[0197] The differences in the external mirror systems 90 b and 90 leadto modifications of Eqs. (38) such that the magnitude of phase shiftsφ_(3b) and φ_(4b), counterparts to phase shifts φ₃ and φ₄, respectively,are related to the round-trip physical length L_(bi) of path i ofmeasurement path 98 b and to reference paths as shown in FIGS. 4b and 4c according to the formulae $\begin{matrix}{{\phi_{1b} = {{\sum\limits_{i = 1}^{i = p}{L_{bi}k_{1}}} + \zeta_{1b}}},{\phi_{2b} = {{\sum\limits_{i = 1}^{i = p}{L_{bi}k_{2}}} + {\zeta_{2b}.}}}} & (52)\end{matrix}$

[0198] For those applications where changes in the measurement path canbe measured interferometrically, a feature for example of an applicationbased on a distance measuring interferometer employed for measuringchanges in the measurement path (c.f. the second preferred embodiment),the phase redundancy in θ can be resolved by recording the change in θas the movable mirror 92 of the external mirror system 90 is scanned ina controlled manner by translator 67 over a given length from a nullposition, the null position being the position where the physicallengths of the measurement and reference legs are the substantially thesame. The accuracy required for the determination of the null positionis typically less accurate than the accuracy required for otherparameters as exemplified in the following example: for λ₁=0.633 μm,(n₁−1)≅3×10⁻⁴, (n₂−n₁)≅1×10⁻⁵, ε≅10⁻⁹, and the condition set fourth inEq. (17), the desired accuracy for the null position determinationcorresponds to an uncertainty in θ of the order of ±3.

[0199] For those applications where the determination of therefractivity and/or or the change in the optical path length due to thegas in a measurement leg is made and mirror 92 of the external mirrorsystem does not have a scanning capability such as considered in thepreceding paragraph, other procedures are available for the resolutionof the phase redundancies of θ and θ_(b). The effective wavelengths of θand θ_(b) are substantially the same so that only procedures for theresolution of phase redundancy in either θ or θ_(b) need be described.

[0200] The second procedure described for the resolution of the phaseredundancy of Φ in the description of the second embodiment can beadapted for the resolution of the phase redundancies of θ_(b), thesecond procedure being based on the use of a series of external mirrorsystems of type 90 b where the series of round-trip physical lengths ofthe series of external mirror systems form a geometric progression. Thesmallest or first round-trip physical length in the series of round-tripphysical lengths will be approximately λ₁/8 divided by the relativeprecision that the initial value of θ_(b) is known. The secondround-trip physical length in the series of round-trip physical lengthswill be approximately the length of the first round-trip physical lengthin the series of round-trip physical lengths divided by the relativeprecision that θ_(b) is measured using the first round-trip physicallength in the series of round-trip physical lengths. This is again ageometric progression procedure, the resulting series of round-tripphysical lengths forming a geometric progression, which is continueduntil the physical length of the external mirror system 90 b used tomeasure the refractivity or the change in optical path length due to therefractivity of the gas would be exceeded if the number in the series ofround-trip physical lengths were incremented by one. For the resolutionof phase redundancy in θ_(b), a typical round-trip physical length forthe first round-trip physical length in the series of round-tripphysical lengths is of the order of 0.5 mm, a typical round-tripphysical length for the second round-trip physical length in the seriesof round-trip physical lengths is of the order of 50 mm, and a typicalround-trip physical length for a third round-trip physical length in theseries of round-trip physical lengths if required is of the order of5000 mm. The physical lengths in the series of physical lengths used forthe resolution of phase redundancy in Φ_(b) are typically orders ofmagnitude larger than the physical lengths in the series of physicallengths used for the resolution of phase redundancy in θ_(b).

[0201] A third procedure is based upon the use of a source (not shown inFIGS. 4a-4 c) of a series of known wavelengths and measuring θ_(b) forthese wavelengths. The number of known wavelengths required for theresolution of the phase redundancy is generally comprised of a smallset.

[0202] Another procedure to resolve the phase redundancy in θ_(b) is toobserve the changes in θ_(b) as the measuring path 98 b is changed fromgas to an evacuated state (the vacuum pump and requisite gas handlingsystem are not shown in FIGS. 4a-4 c). The problems normally encounteredin measuring absolute values for refractivity and changes in the opticalpath length due to the refractivity of the gas based in part on changingthe gas pressure from a non-zero value to a vacuum are not present inthe third preferred embodiment because of a relatively large uncertaintyof the order of ±3 typically permitted in the determination of θ_(b).

[0203] The offset terms ξ_(b) and Z_(b) that are present in Eq. (51) anddefined in Eqs. (47) and (48), respectively, are terms that may requiredetermination and may require monitoring if variable in time. Oneprocedure for the determination of ξ_(b) and Z_(b) is based onreplacement of mirror 91 b of the external mirror system 90 b with amirror Z91 b (not shown in FIGS. 4a-4 c) having a surface Z93 bcorresponding to surface 93 b of mirror 91 b coated so as be areflecting surface for both wavelengths λ₁ and λ₂ and measuring theresulting θ_(b) and Φ_(b). Let the resulting values of θ_(b) and Φ_(b)be θ_(bR) and Φ_(bR), respectively. The quantities ξ_(b) and Z_(b) arerelated to θ_(bR) and Φ_(bR), respectively, as evident from Eqs. (47)and (48) by the formulae

ξ_(b)=θ_(bR),  (53)

Z _(b)=Φ_(bR).  (54)

[0204] The non-electronic contributions to ξ_(b) and Z_(b) should besubstantially constant in time because of the significant level ofcompensation that takes place in the differential plane mirrorinterferometers 69 b and 70 b, beam splitter 65 b, and external mirrorsystem 90 b. The electronic contributions to ξ_(b) and Z_(b) aremonitored by purely electronic means (not shown).

[0205] The wavenumber χ is calculated by the computer using Eq. (47) andthe measured values for θ_(b) and ξ_(b). The ratio K/χ is calculated bythe computer using Eq. (51).

[0206] The refractivity of a gas and/or the change in the optical pathlength of a measurement path due to the gas is subsequently obtainedusing Eqs. (7) and/or (14), respectively. The remainder of thedescription of the fourth preferred embodiment is the same as that givenfor corresponding portions of the second and third preferredembodiments.

[0207] There are three variants of the fourth embodiment wherein thedescription of each of the three variants of the fourth embodiment isthe same as the description given for corresponding portions of thethree variants of the second preferred embodiment.

[0208] It will be appreciated by those skilled in the art that thewavelength λ₁ of the light beam used for the determination of φ₁ in Eqs.(14) and (20) may be different from both of the two wavelengths used todetermine the change in the optical path length of the measuring pathdue to gas in the measuring path without departing from the scope andspirit of the present invention. The requisite reciprocal dispersivepower Γ₃ would be defined in terms of the indices of refraction n₁, n₂,and n₃ of the gas at the three wavelengths λ₁, λ₂, and λ₃, respectively,according to the formula $\begin{matrix}{\Gamma_{3} = \frac{\left( {n_{1} - 1} \right)}{\left( {n_{3} - n_{2}} \right)}} & (55)\end{matrix}$

[0209] for λ₃<λ₂.

[0210] It will be further appreciated by those skilled in the art thatthe two frequency components of either or both beams 9 and 10 may bespatially separated at any point following the means for introducing thefrequency shifts and prior to entering the respective interferometers ofthe described preferred embodiments without departing from the scope andspirit of the present invention. If the two frequency components of theeither of the two beams are spatially separated for any significantdistance from the respective interferometer, it may be necessary toemploy alternative reference beams such as described in the firstembodiment.

[0211] It will also be appreciated by those skilled in the art that thedifferential plane mirror interferometer and the external mirror systemof the additional means for the determination of χ and/or the ratio(K/χ) in the fourth preferred embodiment may be configured such that oneof the light beams corresponding to one of the wavelengths may enter andexit from one end of the external mirror system and a second of thelight beams corresponding to a differing second wavelength may enter andexit from an opposite end of the external mirror system in contrast tothe same end as disclosed in the fourth preferred embodiment withoutdeparting from the scope or spirit of the invention as defined in theclaims. With the reconfiguring of the external mirror system, beamsplitter 65 b may obviously be omitted, the light beams of differingwavelengths entering and exiting through the mirrors 91 b and 92 b withthe reflecting and transmitting coatings on mirror surfaces 95 b and 96b having been reconfigured accordingly.

[0212] The illustrations in FIGS. 2a-2 e and 4 a-4 c depict twopreferred embodiments of the present invention wherein all of theoptical beams for an embodiment are in a single plane. Clearly,modifications using multiple planes can be made to one or more of thetwo preferred embodiments and variants thereof without departing fromthe scope and spirit of the invention.

[0213] The second and fourth preferred embodiments of the presentinvention have external mirror systems 90 b and/or 90 wherein themeasurement paths for λ₁ and λ₂ have the same round-trip physicallengths and the reference paths for λ₁ and λ₂ have the same round-tripphysical lengths. It will be appreciated by those skilled in the artthat the measurement paths for λ₁ and λ₂ can have different physicallengths and the reference paths for λ₁ and λ₂ can have differentphysical lengths without departing from the scope and spirit of thepresent invention as defined in the claims. It will be furtherappreciated by those skilled in the art that the measurement paths forλ₁ and λ₂ can be physically displaced one from the other and thereference paths for λ₁ and λ₂ can be physically displaced one from theother without departing from the scope and spirit of the presentinvention as defined in the claims although there may be somedegradation in performance with regard frequency response of theembodiments and/or in accuracy of calculated quantities due to forexample spatial gradients in the refractivity of a gas in a measurementpath.

[0214] It will be appreciated by those skilled in the art thatalternative data processing may be considered for the four preferredembodiments and variants thereof of the present invention withoutdeparting from the spirit and scope of the present invention.

[0215] The four preferred embodiments and variants thereof of thepresent invention are all configured for use of heterodyne detection. Itwill be appreciated by those skilled in the art that homodyne detectioncan be employed in each of the four preferred embodiments and variantsthereof without departing from the scope and spirit of the presentinvention as defined in the claims. Homodyne receivers would be employedsuch as disclosed in commonly owned U.S. Pat. No. 5,663,793 entitled“Homodyne Interferometric Receiver and Method,” issued Sep. 2, 1997 inthe name of P. de Groot. The computation of the refractivity of a gasand/or the change in the optical path length of a measurement path dueto the gas would be obtained for example in the homodyne version of thefirst preferred embodiment directly from homodyne phases φ_(1H) andφ_(2H), the homodyne phases φ_(1H) and φ_(2H) being counterparts tophases φ₁ and φ₂ of the first preferred embodiment, and with homodyneversions of Eqs. (7) and (14).

[0216] The third and fourth preferred embodiments of the presentinvention measure the ratio (K/χ) and/or χ and use the measured valuesof (K/χ) and/or χ in the computation of the refractivity of a gas and/orthe change in the optical path length of a measurement path due to thegas. It will be appreciated by those skilled in the art that themeasured values of (K/χ) and/or χ can be used as error signals in afeedback system such the condition expressed by Eq. (18) is satisfiedand/or such that χ is constant without departing from the scope andspirit of the present invention as defined in the claims. The measuredvalue of (K/χ) and/or χ in the feedback system are sent to either source1 and/or source 2 and used to control the respective wavelengths ofeither source 1 and/or source 2, for example by controlling theinjection current and/or temperature of a diode laser or the cavityfrequency of an external cavity diode laser.

[0217] It will be appreciated by those skilled in the art thatcombinations of the means of the third and fourth preferred embodimentsto measure the ratio (K/χ) and/or χ and of the means of the first andsecond preferred embodiments may be used to determine the refractivityof a gas and/or the change in the optical path length of a measurementpath due to the gas other than the combinations used in the third andfourth preferred embodiments without departing from the scope or spiritof the invention as defined in the claims.

[0218] Reference is now made to FIG. 5 which is a generalized flowchartdepicting via blocks 500-526 various steps for practicing an inventivemethod for measuring and monitoring the refractivity of a gas in ameasurement path and/or the change in the optical path length of themeasurement path due to the gas wherein the refractivity of the gas maybe changing and/or the physical length of the measurement path may bechanging. While it will be evident that the inventive method depicted inFIG. 5 may be carried out using the inventive apparatus disclosedhereinabove, it will also be apparent to those skilled in the art thatit may also be implemented with apparatus other than that disclosed. Forexample, it will be apparent that one need not use differential planemirror interferometers such as that used in the preferred embodiments,but rather may use other conventional interferometric arrangements solong as the required reference and measurement legs are present. Inaddition, it will be evident that one may use either a homodyne approachor one in which heterodyning techniques are advantageously employed. Aswill be further appreciated, many of the steps in FIG. 5 may be carriedout via appropriate software run on a general purpose computer or asuitably programmed microprocessor either of which may be used tocontrol other elements of the system as needed.

[0219] As seen in FIG. 5, one starts in block 500 by providing two ormore light beams having different wavelengths which preferably have anapproximate approximate relationship as previously described. In block502, the light beams are separated into components which in block 504are preferably altered by either polarization or spatial encoding, orfrequency shifting or both. Otherwise, the light beams may simply beleft unaltered and passed through to block 506.

[0220] As shown in blocks 522 and 524, the relationship of thewavelengths of the light beams may be monitored and if their wavelengthsare not within the limits previously discussed, one can adopt correctivemeasures to compensate from departures of the relationship of thewavelengths from the desired relationship of the wavelengths. Either thedepartures can be used to provide feedback to control the wavelengths ofthe light beam sources or corrections can be established and used insubsequent calculations which are influenced by departures or somecombination of both approaches can be implemented.

[0221] In parallel or contemporaneously with generating the light beamsin block 500, one also provides as indicated in block 526 aninterferometer having two legs, a reference leg and the other ameasurement leg wherein a portion of the measurement path is in a gaswhose refractivity and/or effect on the optical path length of themeasurement path are to be measured.

[0222] As shown by blocks 506 and 508, the previously generated lightbeam components are introduced into the interferometer legs so that eachcomponent has its phase shifted based on the optical path length itexperiences in traveling through the physical length of its assignedleg.

[0223] After the beams emerge from block 508, they are combined in block510 to generate a mixed optical signal. These mixed optical signals arethen sent to block 512 where by means of photodetection correspondingelectrical signals, preferably heterodyne, are generated, and theseelectrical signals contain information about the relative phases betweenthe light beam components. Preferably the electrical signals areheterodyne signals brought about by previously frequency shiftingtreatment.

[0224] In block 514, the electrical signals may be directly analyzed toextract relative phase information which can then be passed on to blocks516-520 or, superheterodyne signals are generated, or modifiedheterodyne and then superheterodyne signals, or modified heterodynesignals, which are then subsequently analyzed for the relative phaseinformation.

[0225] In block 516, any phase ambiguities in homodyne, heterodyne,and/or superheterodyne signals are resolved, preferably by means andcalculations previously elaborated in connection with describing thepreferred embodiments.

[0226] In block 518, the refractivity of the gas and/or the effect ofthe refractivity of the gas on the optical path length of themeasurement path are calculated, corrections are applied as previouslydescribed, and output signals are generated for subsequent downstreamapplications or data format requirements.

[0227] Those skilled in the art may make other changes to the inventiveapparatus and methods without departing from the scope of the inventiveteachings. Therefore, it is intended that the embodiments shown anddescribed be considered as illustrative and not in a limiting sense.

[0228] The interferometry systems described above can be especiallyuseful in lithography applications (as, for example, represented at 267)used for fabricating large scale integrated circuits such as computerchips and the like. Lithography is the key technology driver for thesemiconductor manufacturing industry. Overlay improvement is one of thefive most difficult challenges down to and below 100 nm line widths(design rules), see for example the Semiconductor Industry Roadmap, p82(1997). Overlay depends directly on the performance, i.e. accuracy andprecision, of the distance measuring interferometers used to positionthe wafer and reticle (or mask) stages. Since a lithography tool mayproduce $50-100 M/year of product, the economic value from improvedperformance distance measuring interferometers is substantial. Each 1%increase in yield of the lithography tool results in approximately $1M/year economic benefit to the integrated circuit manufacturer andsubstantial competitive advantage to the lithography tool vendor.

[0229] The function of a lithography tool is to direct spatiallypatterned radiation onto a photoresist-coated wafer. The processinvolves determining which location of the wafer is to receive theradiation (alignment) and applying the radiation to the photoresist atthat location (exposure).

[0230] To properly position the wafer, the wafer includes alignmentmarks on the wafer that can be measured by dedicated sensors. Themeasured positions of the alignment marks define the location of thewafer within the tool. This information, along with a specification ofthe desired patterning of the wafer surface, guides the alignment of thewafer relative to the spatially patterned radiation. Based on suchinformation, a translatable stage supporting the photoresist-coatedwafer moves the wafer such that the radiation will expose the correctlocation of the wafer.

[0231] During exposure, a radiation source illuminates a patternedreticle, which scatters the radiation to produce the spatially patternedradiation. The reticle is also referred to as a mask, and these termsare used interchangeably below. In the case of reduction lithography, areduction lens collects the scattered radiation and forms a reducedimage of the reticle pattern. Alternatively, in the case of proximityprinting, the scattered radiation propagates a small distance (typicallyon the order of microns) before contacting the wafer to produce a 1:1image of the reticle pattern. The radiation initiates photo-chemicalprocesses in the photoresist that convert the radiation pattern into alatent image within the photoresist.

[0232] The interferometry systems described above are importantcomponents of the positioning mechanisms that control the position ofthe wafer and reticle, and register the reticle image on the wafer.

[0233] In general, the lithography system, also referred to as anexposure system, typically includes an illumination system and a waferpositioning system. The illumination system includes a radiation sourcefor providing radiation such as ultraviolet, visible, x-ray, electron,or ion radiation, and a reticle or mask for imparting the pattern to theradiation, thereby generating the spatially patterned radiation. Inaddition, for the case of reduction lithography, the illumination systemcan include a lens assembly for imaging the spatially patternedradiation onto the wafer. The imaged radiation exposes photoresistcoated onto the wafer. The illumination system also includes a maskstage for supporting the mask and a positioning system for adjusting theposition of the mask stage relative to the radiation directed throughthe mask. The wafer positioning system includes a wafer stage forsupporting the wafer and a positioning system for adjusting the positionof the wafer stage relative to the imaged radiation. Fabrication ofintegrated circuits can include multiple exposing steps. For a generalreference on lithography, see, for example, J. R. Sheats and B. W.Smith, in Microlithography: Science and Technology (Marcel Dekker, Inc.,New York, 1998), the contents of which are incorporated herein byreference.

[0234] The interferometry systems described above can be used toprecisely measure the positions of each of the wafer stage and maskstage relative to other components of the exposure system, such as thelens assembly, radiation source, or support structure. In such cases,the interferometry system can be attached to a stationary structure andthe measurement object attached to a movable element such as one of themask and wafer stages. Alternatively, the situation can be reversed,with the interferometry system attached to a movable object and themeasurement object attached to a stationary object.

[0235] More generally, the interferometry systems can be used to measurethe position of any one component of the exposure system relative to anyother component of the exposure system in which the interferometrysystem is attached, or supported by one of the components and themeasurement object is attached, or is supported by the other of thecomponents.

[0236] An example of a lithography scanner 600 using an interferometrysystem 626 is shown in FIG. 6a. The interferometry system is used toprecisely measure the position of a wafer within an exposure system.Here, stage 622 is used to position the wafer relative to an exposurestation. Scanner 600 comprises a frame 602, which carries other supportstructures and various components carried on those structures. Anexposure base 604 has mounted on top of it a lens housing 606 atop ofwhich is mounted a reticle or mask stage 616 used to support a reticleor mask. A positioning system for positioning the mask relative to theexposure station is indicated schematically by element 617. Positioningsystem 617 can include, e.g., piezoelectric transducer elements andcorresponding control electronics. Although, it is not included in thisdescribed embodiment, one or more of the interferometry systemsdescribed above can also be used to precisely measure the position ofthe mask stage as well as other moveable elements whose position must beaccurately monitored in processes for fabricating lithographicstructures (see supra Sheats and Smith Microlithography: Science andTechnology).

[0237] Suspended below exposure base 604 is a support base 613 thatcarries wafer stage 622. Stage 622 includes a plane mirror forreflecting a measurement beam 654 directed to the stage byinterferometry system 626. A positioning system for positioning stage622 relative to interferometry system 626 is indicated schematically byelement 619. Positioning system 619 can include, e.g., piezoelectrictransducer elements and corresponding control electronics. Themeasurement beam reflects back to the interferometry system, which ismounted on exposure base 604. The interferometry system can be any ofthe embodiments described previously.

[0238] During operation, a radiation beam 610, e.g., an ultraviolet (UV)beam from a UV laser (not shown), passes through a beam shaping opticsassembly 612 and travels downward after reflecting from mirror 614.Thereafter, the radiation beam passes through a mask (not shown) carriedby mask stage 616. The mask (not shown) is imaged onto a wafer (notshown) on wafer stage 622 via a lens assembly 608 carried in a lenshousing 606. Base 604 and the various components supported by it areisolated from environmental vibrations by a damping system depicted byspring 620.

[0239] In other embodiments of the lithographic scanner, one or more ofthe interferometry systems described previously can be used to measuredistance along multiple axes and angles associated for example with, butnot limited to, the wafer and reticle (or mask) stages. Also, ratherthan a UV laser beam, other beams can be used to expose the waferincluding, e.g., x-ray beams, electron beams, ion beams, and visibleoptical beams.

[0240] In addition, the lithographic scanner can include a columnreference in which interferometry system 626 directs the reference beamto lens housing 606 or some other structure that directs the radiationbeam rather than a reference path internal to the interferometry system.The interference signal produce by interferometry system 626 whencombining measurement beam 654 reflected from stage 622 and thereference beam reflected from lens housing 606 indicates changes in theposition of the stage relative to the radiation beam. Furthermore, inother embodiments the interferometry system 626 can be positioned tomeasure changes in the position of reticle (or mask) stage 616 or othermovable components of the scanner system. Finally, the interferometrysystems can be used in a similar fashion with lithography systemsinvolving steppers, in addition to, or rather than, scanners.

[0241] As is well known in the art, lithography is a critical part ofmanufacturing methods for making semiconducting devices. For example,U.S. Pat. No. 5,483,343 outlines steps for such manufacturing methods.These steps are described below with reference to FIGS. 6b and 6 c. FIG.6b is a flow chart of the sequence of manufacturing a semiconductordevice such as a semiconductor chip (e.g. IC or LSI), a liquid crystalpanel or a CCD. Step 651 is a design process for designing the circuitof a semiconductor device. Step 652 is a process for manufacturing amask on the basis of the circuit pattern design. Step 653 is a processfor manufacturing a wafer by using a material such as silicon.

[0242] Step 654 is a wafer process which is called a pre-processwherein, by using the so prepared mask and wafer, circuits are formed onthe wafer through lithography. Step 655 is an assembling step, which iscalled a post-process wherein the wafer processed by step 654 is formedinto semiconductor chips. This step includes assembling (dicing andbonding) and packaging (chip sealing). Step 656 is an inspection stepwherein operability check, durability check, and so on of thesemiconductor devices produced by step 655 are carried out. With theseprocesses, semiconductor devices are finished and they are shipped (step657).

[0243]FIG. 6c is a flow chart showing details of the wafer process. Step661 is an oxidation process for oxidizing the surface of a wafer. Step662 is a CVD process for forming an insulating film on the wafersurface. Step 663 is an electrode forming process for forming electrodeson the wafer by vapor deposition. Step 664 is an ion implanting processfor implanting ions to the wafer. Step 665 is a photoresist process forapplying a photoresist (photosensitive material) to the wafer. Step 666is an exposure process for printing, by exposure, the circuit pattern ofthe mask on the wafer through the exposure apparatus described above.Step 667 is a developing process for developing the exposed wafer. Step668 is an etching process for removing portions other than the developedphotoresist image. Step 669 is a photoresist separation process forseparating the photoresist material remaining on the wafer after beingsubjected to the etching process. By repeating these processes, circuitpatterns are formed and superimposed on the wafer.

[0244] The interferometry systems described above can also be used inother applications in which the relative position of an object needs tobe measured precisely. For example, in applications in which a writebeam such as a laser, x-ray, ion, or electron beam, marks a pattern ontoa substrate as either the substrate or beam moves, the interferometrysystems can be used to measure the relative movement between thesubstrate and write beam.

[0245] As an example, a schematic of a beam writing system 700 is shownin FIG. 7. A source 710 generates a write beam 712, and a beam focusingassembly 714 directs the radiation beam to a substrate 716 supported bya movable stage 718. To determine the relative position of the stage, aninterferometry system 720 directs a reference beam 722 to a mirror 724mounted on beam focusing assembly 714 and a measurement beam 726 to amirror 728 mounted on stage 718. Interferometry system 720 can be any ofthe interferometry systems described previously. Changes in the positionmeasured by the interferometry system correspond to changes in therelative position of write beam 712 on substrate 716. Interferometrysystem 720 sends a measurement signal 732 to controller 730 that isindicative of the relative position of write beam 712 on substrate 716.Controller 730 sends an output signal 734 to a base 736 that supportsand positions stage 718. In addition, controller 730 sends a signal 738to source 710 to vary the intensity of, or block, write beam 712 so thatthe write beam contacts the substrate with an intensity sufficient tocause photophysical or photochemical change only at selected positionsof the substrate. Furthermore, in some embodiments, controller 730 cancause beam focusing assembly 714 to scan the write beam over a region ofthe substrate, e.g., using signal 744. As a result, controller 730directs the other components of the system to pattern the substrate. Thepatterning is typically based on an electronic design pattern stored inthe controller. In some applications the write beam patterns aphotoresist coated on the susbstrate and in other applications the writebeam directly patterns, e.g., etches, the substrate.

[0246] An important application of such a system is the fabrication ofmasks and reticles used in the lithography methods described previously.For example, to fabricate a lithography mask an electron beam can beused to pattern a chromium-coated glass substrate. In such cases wherethe write beam is an electron beam, the beam writing system encloses theelectron beam path in a vacuum. Also, in cases where the write beam is,e.g., an electron or ion beam, the beam focusing assembly includeselectric field generators such as quadrapole lenses for focusing anddirecting the charged particles onto the substrate under vacuum. Inother cases where the write beam is a radiation beam, e.g., x-ray, UV,or visible radiation, the beam focusing assembly includes correspondingoptics for focusing and directing the radiation to the substrate.

[0247] Yet other changes may be made to the invention. For example, itmay be desirable in certain applications to monitor the refractive indexof the gas contained on both the reference and in the measurement legsof the interferometer. Examples include the well-known column referencestyle of interferometer, in which the reference leg comprises a targetoptic placed at one position within a mechanical system, and themeasurement leg comprises a target optic placed at a different positionwithin the same mechanical system. Another example application relatesto the measurement of small angles, for which both the measurement andreference beams impinge upon the same target optic but at a smallphysical offset, thereby providing a sensitive measure of the angularorientation of the target optic. These applications and configurationsare well known to those skilled in the art and the necessarymodifications are intended to be within the scope of the invention.

[0248] Additional alternative means of achieving substantialinsensitivity to Doppler shifting in a heterodyne interferometer is totrack the Doppler shift and compensate by either (1) adjusting thefrequency difference between the reference and measurement beams, (2)adjusting the clock frequency of one or both of the electronic A/Dmodules or (3) any similar means of continuously matching the apparentheterodyne beat frequency of the two wavelengths by active adjustment ofthe drive or detection electronics.

[0249] It is understood that while the invention has been described inconjunction with the detailed description thereof, the foregoingdescription is intended to illustrate and not limit the scope of theinvention, which is defined by the scope of the appended claims.

What is claimed is:
 1. Interferometric apparatus for measuring theeffects of the refractive index of a gas in a measurement path, saidinterferometric apparatus comprising: interferometer means comprisingfirst and second measurement legs, said first and second measurementlegs having optical paths structured and arranged such that at least oneof them has a variable physical length and at least one of them is atleast in part occupied by the gas, the optical path length differencebetween said first and second measurement legs varying in accordancewith the difference between the respective physical lengths of theiroptical paths and the properties of said gas; means for generating atleast two light beams having different wavelengths; means forintroducing first and second predetermined portions of each of saidlight beams into said first and second measurement legs, respectively,of said interferometer means so that each of at least one of said firstand second predetermined portions of said light beams travels throughsaid first and second measurement legs along predetermined optical pathswith the same number of passes, said predetermined first and secondportions of said light beams emerging from said interferometer means asexit beams containing information about the respective optical pathlengths through said first and second measurement legs at saidwavelengths; means for combining said exit beams to produce mixedoptical signals containing information corresponding to the phasedifferences between each of said exit beams from corresponding ones ofsaid predetermined optical paths of said first and second measurementlegs at said wavelengths; means for detecting said mixed optical signalsand generating electrical interference signals containing informationcorresponding to the effects of the index of refraction of the gas atsaid different beam wavelengths and the difference in physical pathlengths of said measurement legs and their relative rate of change; andelectronic means for analyzing said electrical interference signals todetermine the effects of said gas in said measurement leg(s) whilecompensating for the relative rates at which the physical path lengthsof said first and second measurement legs are changing.
 2. Theinterferometric apparatus of claim 1 wherein said electronic meanscomprises means for directly receiving said electrical interferencesignals and detecting phases therefrom to generate phase signals wheresaid phase signals contain information corresponding to the effects ofthe index of refraction of the gas at said different beam wavelengthsand the difference in physical path lengths of said measurement legs andtheir rates of change.
 3. The interferometric apparatus of claim 2wherein said electronic means further comprises means for resolvingphase redundancy in said phase signals.
 4. The interferometric apparatusof claim 1 wherein said different wavelengths have an approximateharmonic relationship to each other, said approximate harmonicrelationship being expressed as a sequence of ratios, each ratio beingcomprised of a ratio of low order, non-zero integers.
 5. Theinterferometric apparatus of claim 1 wherein said electronic meansfurther includes phase analyzing means for receiving said electricalinterference signals and generating initial electrical phase signalscontaining information corresponding to the effects of the index ofrefraction of the gas at said different beam wavelengths and thephysical path lengths of said measurement legs occupied by said gas andtheir rates of change.
 6. The interferometric apparatus of claim 5wherein said electronic means further includes multiplying means formultiplying said initial phase signals by factors proportional to saidwavelengths to generate modified phase signals.
 7. The interferometricapparatus of claim 6 wherein said electronic means further includesmeans for receiving said modified phase signals and selectively addingand subtracting them to generate sum and difference phase signalscontaining information corresponding to the effects of the index ofrefraction of the gas at said different beam wavelengths and thedifference in physical path lengths of said measurement legs o and theirrelative rates of change.
 8. The interferometric apparatus of claim 7wherein said electronic means further includes means for receiving saidsum and difference phase signals and at least one of said initial phasesignals to determine the difference in physical lengths, L, of saidmeasurement legs.
 9. The interferometric apparatus of claim 8 furtherincluding means for resolving redundancies among said initial phase andsaid sum and difference phase signals.
 10. The interferometric apparatusof claim 6 wherein said wavelengths are non-harmonically related. 11.The interferometric apparatus of claim 6 wherein said wavelengths ofsaid light beams have an approximate harmonic relationship to eachother, said approximate harmonic relationship being expressed as asequence of ratios, each ratio being comprised of a ratio of low ordernon-zero integers.
 12. The interferometric apparatus of claim 1 furtherincluding a microlithographic means operatively associated with saidinterferometric apparatus for fabricating integrated circuits on wafers,said microlithographic means comprising: at least one stage; anillumination system for imaging spatially patterned radiation onto thewafer; and at least one positioning system for adjusting the position ofsaid at least one stage; wherein said interferometric apparatus isadapted to measure the position of said at least one stage.
 13. Theinterferometric apparatus of claim 1 further including amicrolithographic means operatively associated with said interferometricapparatus for use in fabricating integrated circuits on a wafer, saidmicrolithographic means comprising: at least one stage for supporting awafer; an illumination system including a radiation source, a mask, apositioning system, a lens assembly, and predetermined portions of saidinterferometric apparatus, said microlithographic means being operativesuch that the source directs radiation through said mask to producespatially patterned radiation, said positioning system adjusts theposition of said mask relative to radiation from said source, said lensassembly images said spatially patterned radiation onto the wafer, andsaid interferometric apparatus measures the position of said maskrelative to said radiation from said source.
 14. The interferometricapparatus of claim 1 further including microlithographic apparatusoperatively associated with said interferometric apparatus forfabricating integrated circuits comprising first and second components,said first and second components being moveable relative to one another,said first first and second components being connected with said firstand second measurement legs, moving in concert therewith, such that saidinterferometric apparatus measures the position of said first componentrelative to said second component.
 15. The interferometric apparatus ofclaim 1 further including a beam writing system operatively associatedwith said interferometric apparatus for use in fabricating a lithographymask, said beam writing system comprising: a source for providing awrite beam to pattern a substrate; at least one stage for supporting asubstrate; a beam directing assembly for delivering said write beam tothe substrate; and a positioning system for positioning said at leastone stage and said beam directing assembly relative to one another, saidinterferometric apparatus being adapted to measure the position of saidat least one stage relative to said beam directing assembly.
 16. Theinterferometric apparatus of claim 11 wherein the relative precision ofsaid approximate harmonic relationship expressed as said sequence ofratios is an order of magnitude or more less than the dispersion of therefractive index of said gas times the relative precision, ε, requiredfor the measurement of the refractivity of said gas or of the change inthe difference in optical path lengths of said measurement legs.
 17. Theinterferometric apparatus of claim 1 wherein the relative precision ofthe relationship between said wavelengths expressed as a ratio(s)relative to a known ratio is greater than a predetermined valuecorresponding to the precision requirements of a downstream application.18. The interferometric apparatus of claim 17 further including meansfor monitoring the relative precision of said relationship expressed assaid ratio.
 19. The interferometric apparatus of claim 18 furtherincluding means responsive to said means for monitoring said relativeprecision of said relationship expressed as said ratio for providing afeedback signal to control said means for generating said light beams sothat said relative precision of said relationship expressed as saidratio is substantially equal to or less than the predetermined valuecorresponding to the precision requirements of a downstream application.20. The interferometric apparatus of claim 1 wherein said at least twolight beams each have orthogonal polarization states.
 21. Theinterferometric apparatus of claim 20 further including means forintroducing a frequency difference(s) between said orthogonalpolarization states of said light beams.
 22. The interferometricapparatus of claim 21 wherein said means for detecting said mixedoptical signals comprises a single photodetector for receiving selectedones of said mixed optical signals beams having predetermined frequencydifferences.
 23. The interferometric apparatus of claim 21 wherein saidmeans for combining said exit beams are adapted to mix said polarizationstates of said light beams.
 24. The interferometric apparatus of claim23 wherein said information corresponding to said phase differences insaid mixed optical signals are phase shifts φ_(j) related to thedifferences in total round-trip physical lengths pL of said measurementlegs occupied by said gas according to the formulae φ_(j) =Lpk _(j) n_(j)+ζ_(j), j=1, 2, . . . where k _(j)=2π/λ_(j), p is the number ofmultiple passes through the at least one of said first and secondmeasurement legs, and n_(j) are the refractive indices of gas in said atleast one of said measurement legs occupied by the gas corresponding towavelength, λ_(j), and ζ_(j) are phase offsets not associated withoptical paths.
 25. The interferometric apparatus of claim 24 whereinsaid electrical interference signals comprise heterodyne signals of theform: s _(j) =A _(j)cos[α_(j)(t)], j=1, 2, . . . where thetime-dependent arguments α_(j)(t) are given by α_(j)(t)=2πf _(j) t+φ_(j), j=1, 2 . . . .
 26. The interferometric apparatus of claim 25wherein said different wavelengths have an approximate harmonicrelationship to each other, said approximate harmonic relationship beingexpressed as a sequence of ratios, each ratio being comprised of a ratioof low order, non-zero integers.
 27. The interferometric apparatus ofclaim 1 wherein said different wavelengths have a relationship to eachother expressed as a ratio, said ratio being non-harmonic.
 28. Theinterferometric apparatus of claim 25 wherein said electronic means isadapted to receive said heterodyne signals and determine said phaseshifts, φ_(j) =Lpk _(j) n _(j)+ζ_(j), j=1, 2, . . . .
 29. Theinterferometric apparatus of claim 26 further including means forgenerating at least two modified heterodyne signals from said heterodynesignals, said modified heterodyne signals being of the form: {tilde over(s)} _(j) =Ã _(j)cos[p _(j)α_(j)(t)], j=1, 2, . . . , where:α_(j)(t)=2πf _(j) t+φ _(j), j=1, 2, . . . . where p_(j) are non-zerointegers where p₁≠p₂ and have the same ratio as that of said approximateharmonic relationship of said wavelengths.
 30. The interferometricapparatus of claim 29 further including means for generating asuperheterodyne signal of the form: S={tilde over (s)} ₁ {tilde over(s)} ₂ where said superheterodyne signal S is comprised of two sidebandswith a suppressed carrier expressed as: S=S ⁺ +S ⁻ where S ⁺=½Ã ₁ Ã₂cos(2πνt+θ),${S^{-} = {\frac{1}{2}{\overset{\sim}{A}}_{1}{\overset{\sim}{A}}_{2}{\cos \left( {{2\pi \quad {Ft}} + \Phi} \right)}}},$

ν=(p ₁ f ₁ +p ₂ f ₂) θ=(p ₁φ₁ +p ₂φ₂) F=(p ₁ f ₁ −p ₂ f ₂) Φ=(p ₁φ₁ −p₂φ₂), whereby said superheterodyne signal S is comprised of twosidebands, S⁺ and S⁻, of equal amplitude, one sideband with frequency νand phase θ and a second sideband with frequency F and phase Φ.
 31. Theinterferometric apparatus of claim 26 further including means forgenerating at least two modified heterodyne signals from said heterodynesignals, said modified heterodyne signals being of the form:${{\overset{\sim}{s}}_{1}^{\prime} = {{\overset{\sim}{A}}_{1}^{\prime}{\cos \left\lbrack {{\alpha_{1}(t)}/p_{2}} \right\rbrack}}},{{\overset{\sim}{s}}_{2}^{\prime} = {{\overset{\sim}{A}}_{2}^{\prime}{\cos \left\lbrack {{\alpha_{2}(t)}/p_{1}} \right\rbrack}}}$

where α_(j)(t)=2πf _(j) t+φ _(j), j=1, 2, . . . and p₁ and p₂ arenon-zero integers with p₁≠p₂ and have the same ratio as that of saidapproximate harmonic relationship of said wavelengths.
 32. Theinterferometric apparatus of claim 1 wherein said electronic means isfurther adapted to determine the difference in physical lengths, pL, ofsaid measurement legs occupied by said gas.
 33. The apparatus of claim32 wherein said electronic means is configured to receive the intrinsicoptical property, the reciprocal dispersive power, Γ, of the gas tocalculate the difference in physical lengths, L, as:${\Gamma = \frac{\left\lbrack {{n_{1}\left( \lambda_{1} \right)} - 1} \right\rbrack}{\left\lbrack {{n_{3}\left( \lambda_{3} \right)} - {n_{2}\left( \lambda_{2} \right)}} \right\rbrack}},{and}$

λ₁, λ₂, and λ₃ are wavelengths and n₁, n₂, and n₃ are indices ofrefraction and wherein the denominator may be replaced by [n ₃(λ₃)−n₁(λ₁)] or [n ₂(λ₂)−n ₁(λ₁)].
 34. The interferometric apparatus of claim1 further including a microlithographic means operatively associatedwith said interferometer means such that said difference in physicallengths, L, may be used to determine the change in relative distancebetween predetermined elements of said microlithographic means.
 35. Theinterferometric apparatus of claim 1 further including means foraligning said light beams into a single substantially collinear lightbeam that travels along said at least one of said measurement pathsoccupied by gas.
 36. The interferometric apparatus of claim 1 whereinsaid interferometer means comprises a polarized Michelsoninterferometer.
 37. The interferometric apparatus of claim 1 whereinsaid interferometer means comprises a differential plane mirrorinterferometer.
 38. The interferometer apparatus of claim 1 furtherincluding means for directly receiving said electrical interferencesignals from said means for detecting and converting said electricalinterference signals to digital form to reduce phase errors in furtherdownstream calculations.
 39. Interferometric method for measuring theeffects of the refractive index of a gas in a measurement path, saidinterferometric apparatus comprising: providing an interferometer meanscomprising a first and second measurement legs, said first and secondmeasurement legs having optical paths structured and arranged such thatat least one of them has a variable physical length and at least one ofthem is at least in part occupied by gas, the optical path lengthdifference between said first and second measurement legs varying inaccordance with the difference in the respective physical lengths oftheir optical paths and the properties of said gas; generating at leasttwo light beams having different wavelengths; introducing first andsecond predetermined portions of each of said light beams into saidfirst and second measurement legs, respectively, of said interferometermeans so that each of at least one of said first and secondpredetermined portions of said light beams travels through said firstand second measurement legs along predetermined optical paths with thesame number of passes, said predetermined first and second portions ofsaid light beams emerging from said interferometer means as exit beamscontaining information about the respective optical path lengths throughsaid first and second measurement legs at said wavelengths; combiningsaid exit beams to produce mixed optical signals containing informationcorresponding to the phase differences between each of said exit beamsfrom corresponding ones of said predetermined optical paths of saidfirst and second measurement legs at said wavelengths; detecting saidmixed optical signals and generating electrical interference signalscontaining information corresponding to the effects of the index ofrefraction of the gas at said different beam wavelengths and therelative physical path lengths between said first and second measurementlegs and their relative rates of change; and electronically analyzingsaid interference electrical signals to determine the effects of saidgas in said measurement leg(s) while compensating for the relative ratesat which the physical path lengths of said first and second measurementlegs are changing.
 40. The interferometric method of claim 39 whereinsaid different wavelengths have an approximate harmonic relationship toeach other, said approximate harmonic relationship being expressed as asequence of ratios, each ratio being comprised of a ratio of low order,non-zero integers.
 41. The interferometric method of claim 40 whereinthe relative precision of said approximate harmonic relationshipexpressed as said sequence of ratios is an order of magnitude or moreless than the dispersion of the refractive index of said gas times therelative precision required for the measurement of the refractivity ofsaid gas or of the change in the differene in optical path lengths ofsaid measurement legs due to said gas.
 42. The interferometric method ofclaim 40 further including the step of monitoring the relative precisionof said approximate harmonic relationship expressed as said sequence ofratios.
 43. The interferometric method of claim 42 further including thestep, responsive to the step of monitoring said relative precision ofsaid approximate harmonic relationship, of providing a feedback signalto control said light beams so that said relative precision of saidapproximate harmonic relationship is within an order of magnitude ormore less than the dispersion of the refractive index of said gas timesthe relative precision required for the measurement of the refractivityof said gas or of the change in the difference in optical path lengthsof said measurement legs due to said gas.
 44. The interferometric methodof claim 39 wherein said at least two light beams each have orthogonalpolarization states.
 45. The interferometric method of claim 44 furtherincluding the step of introducing a frequency difference(s) between saidorthogonal polarization states of said light beams.
 46. Theinterferometric method of claim 45 wherein said step of combining saidexit beams comprises mixing said polarization states of said lightbeams.
 47. The interferometric method of claim 46 wherein saidinformation corresponding to said phase differences in said mixedoptical signals are phase shifts φ_(j) related to the differences intotal round-trip physical lengths pL of said measurement legs occupiedby said gas according to the formulae φ_(j) =Lpk _(k) n _(j)+ζ_(j), j=1,2, . . . where k _(j)=2π/λ_(j), p is the number of multiple passesthrough at least one of said first and second measurement legs, andn_(j) are the refractive indices of gas in said at least one measurementleg corresponding to wavelength, λ_(j).
 48. The interferometric methodof claim 47 wherein said electrical interference signals compriseheterodyne signals of the form: s _(j) =A _(j)cos[α_(j)(t), j=1, 2, . .. where the time-dependent arguments α_(j)(t) are given by α_(j)(t)=2πf_(j) t+φ _(j), j=1, 2, . . . .
 49. The interferometric method of claim48 wherein said step of electronically analyzing comprises receivingsaid heterodyne signals and determining said phase shifts, φ_(j) =Lpk_(j) n _(j)+ζ_(j), j=1, 2, . . .
 50. The interferometric method of claim48 wherein said wavelengths are harmonically related and furtherincluding the step of generating at least two modified heterodynesignals from said heterodyne signals, said modified heterodyne signalsbeing of the form: {tilde over (s)} _(j) =Ã _(j)cos[p _(j)α_(j)(t)],j=1, 2, . . . where: α_(j)(t)=2πf _(j) t+φ _(j), j=1, 2, . . . and p₁and p₂ are non-zero integers with p₁≠p₂ and have the same ratio as thatof said approximate harmonic relationship of said wavelengths.
 51. Theinterferometric method of claim 50 further including the step ofgenerating a superheterodyne signal of the form: S={tilde over (s)} ₁{tilde over (s)} ₂ where said superheterodyne signal S is comprised oftwo sidebands with a suppressed carrier expressed as: S=S ⁺ +S ⁻ wherexS⁺=½Ã ₁ Ã ₂cos(2πνt+θ), S ⁻=½Ã₁ Ã ₂cos(2πFt+Φ), ν=(p ₁ f ₁ +p ₂ f ₂)θ=(p ₁φ₁ +p ₂φ₂) F=(p ₁ f ₁ −p ₂ f ₂) Φ=(p ₁φ₁ −p ₂φ₂), whereby saidsuperheterodyne signal S is comprised of two sidebands, S⁺ and S⁻, ofequal amplitude, one sideband with frequency ν and phase θ and a secondsideband with frequency F and phase Φ.
 52. The interferometric method ofclaim 48 further including the step of generating at least two modifiedheterodyne signals from said heterodyne signals, said modifiedheterodyne signals being of the form:${{\overset{\sim}{s}}_{1}^{\prime} = {{\overset{\sim}{A}}_{1}^{\prime}{\cos \left\lbrack {{\alpha_{1}(t)}/p_{2}} \right\rbrack}}},{{\overset{\sim}{s}}_{2}^{\prime} = {{\overset{\sim}{A}}_{2}^{\prime}{\cos \left\lbrack {{\alpha_{2}(t)}/p_{1}} \right\rbrack}}}$

where α_(j)(t)=2πf _(j) t+φ _(j), j=1, 2, . . . and p₁ and p₂ arenon-zero integers with p₁≠p₂ and have the same ratio as that of saidapproximate harmonic relationship of said wavelengths.
 53. Theinterferometric method of claim 39 wherein said step of electronicallyanalyzing comprises determining the difference in physical lengths, L,of said measurement legs.
 54. The interferometric method of claim 53wherein said difference in physical lengths, L, may be used to determinethe relative distance between predetermined elements of amicrolithographic means.
 55. The interferometric method of claim 53wherein said difference in physical lengths, L, is calculated with apredetermined value for the reciprocal relative dispersion, Γ, of saidgas where:${\Gamma = \frac{\left\lbrack {{n_{1}\left( \lambda_{1} \right)} - 1} \right\rbrack}{\left\lbrack {{n_{3}\left( \lambda_{3} \right)} - {n_{2}\left( \lambda_{2} \right)}} \right\rbrack}},{and}$

λ₁, λ₂, and λ₃ are wavelengths and n₁, n₂, and n₃ are indices ofrefraction and wherein the denominator may be replaced by[n₃(λ₃)−n₁(λ₁)] or [n₂(λ₂)−n₁(λ₁)].
 56. The interferometric method ofclaim 39 further including the step of aligning said light beams into asingle substantially collinear light beam that travels along at leastone of said measurement legs.
 57. The interferometric method of claim 39wherein said interferometer means comprises a polarized Michelsoninterferometer.
 58. The interferometric method of claim 39 wherein saidinterferometer means comprises a differential plane mirrorinterferometer.
 59. The interferometer method of claim 39 furtherincluding the step of directly receiving said electrical interferencesignals after their formation to transform them to digital form toreduce phase errors in further downstream calculations.
 60. Theinterferometric method of claim 39 further including the following stepsfor forming integrated circuits on a wafer: providing at least onemoveable stage; imaging spatially patterned radiation onto a wafer;adjusting the position of said at least one stage; and measuring theposition of said at least one stage.
 61. The interferometric method ofclaim 39 further including the steps of: supporting a wafer on at leastone moveable stage; directing a source of radiation through a mask andlens assembly to produce spatially patterned radiation, adjusting theposition of said mask relative to radiation from said source, said lensassembly imaging said spatially patterned radiation onto the wafer, andmeasuring the position of said mask relative to said radiation from saidsource.
 62. The interferometric method of claim 39 further including thestep of providing a microlithographic apparatus for fabricatingintegrated circuits comprising first and second components, said firstand second components being moveable relative to one another, said firstand second components being connected with said first and secondmeasurement legs, respectively, moving in concert therewith, such thatthe position of said first component relative to said second componentis measured.
 63. The interferometric method of claim 39 furtherincluding the steps of: providing a pattern of radiation with a writebeam source; supporting a substrate on at least one stage; directingsaid write beam such that said pattern of radiation impinges onto thesubstrate; and positioning said at least one stage and said write beamof radiation relative to one another, and measuring the position of saidat least one stage relative to said write beam.
 64. The interferometricmethod of claim 39 wherein said wavelengths have a non-harmonicrelationship with respect to one another.
 65. The interferometric methodof claim 39 wherein said step of electronically analyzing furtherincludes the step of receiving said electrical interference signals andextracting the phase therefrom to generate initial electrical phasesignals containing information corresponding to the effects of the indexof refraction of the gas at said different beam wavelengths and thediffernce in physical path lengths of said measurement legs and theirrelative rates of change.
 66. The interferometric method of claim 65wherein said step of electronically analyzing further includes the stepof multiplying said initial phase signals by factors proportional tosaid wavelengths to generate modified phase signals.
 67. Theinterferometric method of claim 66 wherein said step of electronicallyanalyzing further includes the step of receiving said modified phasesignals and selectively adding and subtracting them to generate sum anddifference phase signals containing information corresponding to theeffects of the index of refraction of the gas at said different beamwavelengths and the differences in round trip physical lengths of saidmeasurement legs and their relative rates of change.
 68. Theinterferometric method of claim 67 wherein said step of electronicallyanalyzing further includes the step of receiving said sum and differencephase signals and at least one of said initial phase signals todetermine the difference in round trip physical lengths, L, of saidmeasurement legs.
 69. The interferometric method of claim 67 furtherincluding the step of resolving redundancies among said initial phaseand said sum and difference phase signals.
 70. The interferometricmethod of claim 66 wherein said wavelengths are non-harmonicallyrelated.
 71. The interferometric method of claim 66 wherein saidwavelengths of said light beams have an approximate harmonicrelationship to each other, said approximate harmonic relationship beingexpressed as a sequence of ratios, each ratio being comprised of a ratioof low order non-zero integers.
 72. The interferometric method of claim46 wherein the step of introducing frequency differences between saidorthogonal polarization states of said light beams is such that at leasttwo of said light beams have different frequencies between theirrespective polarization states and so that a single photodetector can beused for generating phase signals from at least two of said exit beams.